So, a bit of a more controversial topic I'm sure, but let's try to have an actual proper conversation here, shall we?
See, I tried to discuss this a couple of times irl before, but whenever someone brings up the lizardfolk/dragonborn/ect with boobs topic, it's nearly always immediately shut down and made fun off because of "lol, no mammal, u so dumb" or "lol, ur just perv" or whatever clever or original comment people come up with. However, I'm not sure I concur with that reasoning.
While it's an undeniable fact, that mammary glands are a thing and tissue swelling during pregnancy in mammals is a thing, it's also an undeniable fact that despite that, permanently enlarged breasts (henceforth referred to as boobs) are exclusively a thing that occurs in humans. For example, at least as far as I'm aware, there aren't any monkey or bears, cats or dogs or pretty much any other mammal species running around, sporting boobs. Even for cows, who are pretty much always used as the poster child for these kinds of arguments, if you ever seen a natural cow, their udders aren't actually really all that big when they are not pumped full with chemical and hormones to keep them in a perpetual pregnancy like state.
So, are "boobs" actually related to being mammalian at all? And if not, what makes humans different, that it only happens in us and not any other species? Well, humans are the only permanently bipedal species on the planet.
It is long since theorized, that "boobs" developed as a natural response to a lack of sexual stimuli within our ancestors. To elaborate, a quadrupedal species constantly has the genitals of their partner right in front of their eyes on eye level, thus giving them enough stimuli to ensure a constant repopulation of their species. However, what if a species suddenly went from being quadrupeds to being bipeds? With the resulting shift in their field of view, they would lack the same stimuli as a quadrupedal species. So, in response, human females evolved permanently enlarged breasts to compensate for that, by having a clearly visible stimuli on eye level.
In that sense, boobs aren't really about being mammal at all, but serve as a necessary evolutionary tool for bipedal species to keep their species from dying out. So it would make complete sense for lizardfolks/dragonborn/ect and other bipedal species to posses boobs for the exact same reason, even if they don't seem to serve as "obvious" a purpose like they do in mammals.
So, to summarize: do boobs on lizardfolks/dragonborn/ect make sense? Yes, as far as I'm concerned, they very much do.
However, I freely admit I'm no expert and there might be things I overlooked, so please, feel free to correct me, if I got anything wrong.

Okay, I'll bite. Yes, breasts have to do with being a mammal. No, not all mammals have breasts, but only mammals have breasts.

However, this is fantasy, and dragonborns / lizardfolk have basically human body shape except for scales and their face. What's more, it's probably pretty hard for humans to distinguish reptilian gender by any other indicator, so if you want people to be able to tell reptilian genders apart, I think it's fine if they have breasts.

There's flat-chested women in the real world and they're still easily recognizable as women, so I don't see why having boobs would be a hard requirement to tell male and female Dragonborn apart. I don't disagree with the "it's fantasy, you can do whatever" thing but I also think giving them mammalian traits kinda defeats the point of having a non-mammalian race and you'd be passing up an opportunity to explore how their social norms might be different. What does a male Dragonborn consider attractive in a woman, and vice-versa? What's considered revealing or sexy clothing in their culture?

Quote from InquisitiveCoder >>

There's flat-chested women in the real world and they're still easily recognizable as women, so I don't see why having boobs would be a hard requirement to tell male and female Dragonborn apart. 

 Those women still have other secondary sex characteristics to distinguish them as female, such as wider hips, facial structure, generally small size, and the lack of an Adam's apple. For anyone unfamiliar with the term, a "secondary sex characteristic" is a biological feature that distinguishes an individual as either male or female while not being directly related to reproduction. One of these characteristics in humans (and unique to humans specifically) is, as mentioned by the OP, enlarged breasts in females. Without such characteristics, it would actually be as difficult to tell that Justin Bieber isn't really a flat chested woman as the internet would have you believe.

Now, dragonborn are specifically not human, and are specifically stated to be biologically related to dragons more than anything else. But, and this is that "big but," they also have humanoid form. Emphasis on humanoid. In any official art I've seen, the only things about dragonborn that I would say resemble dragons more than humans are their skin/scales, heads, and their claws. Visually, they look like a human wearing a particularly good, Hollywood grade costume from an 80's sci-fi movie about lizardmen from space. I'm quite certain they were designed like this (like other playable races) because it's just simpler for rules purposes and with equipment and gear to have all playable characters have the same general humanoid shape. If you're going to give them human form, I say they can have humanoid boobs, though as a DM I would let the player decide that as there's enough lack of specifics from official sources. 

I work under the premise that, in DnD at least, female lizardfolk do not have breasts because they are strictly reptilian.

Dragons, however, are their own clade of creature that resemble that of reptiles, but have distinctly mammalian traits. What do I mean? Wings aside, they have a feline body structure rather than reptilian, with powerful chest muscles to facilitate claw attacks. Their necks are fairly serpentine I will give you, as are their VERY prehensile tails. We also know that their claws, while large, do have a level of dexterity compared to that of humanoids. In fact, I remember a few times where dragons are cited as being able to write (so long as their stylus/pen/whatever are sized for dragons). After all, where do you think Draconic got its written form from?

This being said, dragonborn are (depending on setting) either born of corrupted dragon eggs in a ritual (the Draconians of Krynn) or the creations of the dragon gods (Abeir), or even creations of the dragons themselves (Eberron theoretically). As a result, I don't find it very odd that a purely draconic species would mirror their dragon cousins in being a repto-mammal analogue and, in humanoid form, possess breasts as a female. It would not serve the same purpose as in mammals (being the generation of milk) but rather a food reserve for when times are harsh, or the females are clutching and thus not able to go out and hunt for themselves (outside of civilization). Why on the chest though in a manner resembling that of humans? Natural armor as well. It's harder to get through to vital organs such as the heart, thus making the females more likely to survive an attack.

That said, lizardfolk (and Argonians) definitely should not have breasts on their females because they are purely reptilian, not the chimeric clade that dragons would make up.

So I know this is nitpicky, but being an evolutionary biology nerd almost as much as a fantasy nerd I still feel the need to say that I think the OP forgot about birds (not to mention a number of extinct fauna, notably most non-avian theropod dinosaurs) when mentioning that humans are the only strictly bipedal species on the planet. True, the presence of breasts in humans is probably related to our bipedality, but it also likely wouldn't have evolved IMO if we weren't already primates. So yes, it's still very much related to the fact that we're mammals.

That said, fantasy is fantasy so if players/DMs want to give their lizardfolk ladies cleavage then I don't really have cause for complaint.

Birds have their own secondary sex characteristics for detecting and attracting mates. Most obvious is that male birds of most species have bright plumage to attract the attention of females. Along that line I suppose it would be rational to come up with different shades of color or patterns thereof to differentiate male and female dragonborn, at least if one knows what they're looking at. Or you could just go with the human based gender specific figure, most notably wider hips and boobs for females. Actually, now I want to use both of those. 

DM describes a busty dragonborn woman with alternating swirls of dark and pale blue scale patterns resembling the sky of a Van Gogh painting. 

Dragonborn NPC: I could stare at a beauty like that all day.

Human PC: Yeah, those are some nice knockers.

Dragonborn NPC: I was talking about her scales, but sure, I guess her boobs are nice too. If that's really your thing, I mean.

Quote from Flushmaster>>

Those women still have other secondary sex characteristics to distinguish them as female, such as wider hips, facial structure, generally small size, and the lack of an Adam's apple. For anyone unfamiliar with the term, a "secondary sex characteristic" is a biological feature that distinguishes an individual as either male or female while not being directly related to reproduction. One of these characteristics in humans (and unique to humans specifically) is, as mentioned by the OP, enlarged breasts in females. Without such characteristics, it would actually be as difficult to tell that Justin Bieber isn't really a flat chested woman as the internet would have you believe.

Right. I'm just saying there's no reason to assume they wouldn't have secondary sex characteristics, and even if they didn't, they'd naturally evolve other ways to signal sex and attract mates or arrange relationships (all of which humans do too.) Any species that's survived a long time either has to be good at reproducing or bad at dying. Dragonborn have slightly shorter lifespans than humans (max age of ~80) so they ought to reproduce at least as often as humans do, if not more. That goes double for Lizardfolk. As the saying goes, "Life, uh, finds a way."

I do like the bipedal thing as a somewhat logical explanation for why reptilian creatures would still have breasts. It runs into probably a million contradictions within evolution biology, but so does 90% of fantasy creatures. I think that's more than enough to justify that character design if that's what someone wants for their game.

That said, within the games I DM I avoid giving the various Lizard Races breasts but I still focus on tertiary sexual characteristics. I established that only female Kobolds are ever born winged, and they have longer horns. For Lizardfolk I established that males have the large, colorful head frills while females are on average larger, but lack as many colorful decorations. The players have only encountered Male dragonborn so far, but my intent is that female Dragonborns still have an hourglass figure, but don't necessarily have distinct "breasts". I don't know if everyone will get this reference, but something that comes to mind is the Pokemon "Salazzle"... although be careful about doing a google search on that, since it's attracted some uh... very specific fanart.

Quote from TransmorpherDDS >>

I do like the bipedal thing as a somewhat logical explanation for why reptilian creatures would still have breasts. It runs into probably a million contradictions within evolution biology, but so does 90% of fantasy creatures. I think that's more than enough to justify that character design if that's what someone wants for their game.

That said, within the games I DM I avoid giving the various Lizard Races breasts but I still focus on tertiary sexual characteristics. I established that only female Kobolds are ever born winged, and they have longer horns. For Lizardfolk I established that males have the large, colorful head frills while females are on average larger, but lack as many colorful decorations. The players have only encountered Male dragonborn so far, but my intent is that female Dragonborns still have an hourglass figure, but don't necessarily have distinct "breasts". I don't know if everyone will get this reference, but something that comes to mind is the Pokemon "Salazzle"... although be careful about doing a google search on that, since it's attracted some uh... very specific fanart.

you know, Salazzle is a very good way to maybe do it.

......disregard the incognito window. *slowly closes his computer*

To elaborate, a quadrupedal species constantly has the genitals of their partner right in front of their eyes on eye level, thus giving them enough stimuli to ensure a constant repopulation of their species.

Tell that to giraffes, llamas, alpacas... ;)

There's probably a stronger case that, as a human's senses revolve very heavily on sight and sound, and less so on smells, that we simply needed visual and audible differences (higher voices and breasts) to ensure reproduction as opposed to scent & pheromonal triggers. There's a reason why a female dog in heat attracts the males from all around, and it's nothing to do with her appearance.

I would expect female dragonborn to have egg-bearing hips, and probably a stronger build as they are probably the ones making the nests. I would expect the males to be colourful, as most species involve the males competing for the attention of females and not vice versa.

I feel like breasts are somewhat unnecessary for dragonborn. The only real reason for adding them seems to be to make them more visually relatable to humans, when I would prefer to embrace the fantasy element of the game more.

In older editions, people were converted to Dragonborn. In 5e, they say "born of dragons" which can have different interpretations including being a creation of dragons (similar to older lore) or literally birthed from a dragon. Draconians (a kind of Dragonborn in 5e) are from corrupted eggs. How that corruption was managed isn't specified.

5e also explains that there has been interbreeding with Dragonborn though it is unclear if they mean only between different Dragon-blood lineage or also between different species to arrive at a common appearance rather than strictly appearing like their strongest Draconic bloodline.

So, the answer is: "There's lore to possibly support it if you want it and, as a player, if the DM says it's fine."

As explained a little in this thread, we should really note that when we have humanoid anything that's anything other than a human, you have already gone against anything that makes "anatomical sense". In reality, dragonborn are more mammal than they are reptile. They have reptilian faces, scales, and lay eggs on land and that's pretty much where the similarities end. They aren't cold blooded, they are bipedal, they have advanced emotional and mental faculties, can talk, have butts (nobody seems to have a problem with this which always stuck out to me as a weird antagonism to specifically breasts), have to eat more often, don't shed, lack sensory organs that other reptiles have, and the most defining trait of all, they breath magic (before people say dinosaurs, they are actually a great example of why our preconceived notion of what's "normal in a family" is typically not prescriptive).

It always seems we get caught up in the ", but biology doesn't allow breasts, what would they be used for?" when anyone who studies biology would tell you our categories for animals are extremely subjective. There is literally NOTHING stopping any organism from attaining a body part that another organism has. Bats evolved wings while never having any ancestry of wings, same as insects. Those who paid attention in biology would know this as convergent evolution. I see no biological reason why reptiles can't have breasts. I think it's extremely prescriptive to say that our categorization of families can or cannot have traits, due to our own perspective and otherwise small sample group (only living things from our planet). To say breasts are a "mammalian trait" is to misunderstand how evolution operates. It has nothing to do with being a mammal. It has everything and only to do with adaptation to environment, and we mammals are in the privileged position to have simply gotten them first. If you think that reptiles cannot have breasts from an evolutionary standpoint, I would like to know how you know that. 

Actually, there very much is a reason why a reptile wouldn't evolve breasts.  In order to evolve a feature, an organism must first have the feature that it's evolving from.  Mammary glands are modified sweat glands, which are a trait that mammals evolved before they were actually mammals but reptiles lack.  Reptiles, in general, have significantly fewer glands in their skin than mammals do, and while one of the existing glands in some species of reptile could theoretically eventually evolve into something similar to mammary glands, it wouldn't actually be the same as a mammalian mammary gland- the composition of what it secreted would be quite different.  To get back to the bird/bat comparison, they both have wings which evolved from the front limbs of their non-winged ancestors, but if you actually look at a bird wing vs a bat wing you'll immediately notice that they're extremely different.  A bird's wing is an arm with the hands absent except for some tiny finger bones, while a bat's wing is an arm with an extremely elongated hand and fingers.  Parallel evolution produces similar results, not identical results.

But that's irrelevant because dragonborn didn't evolve naturally over millions of years.  They exist because Bahamut or Tiamat or some powerful mortal dragon said "hey, know what would be cool?  If there were humanoids who looked kinda like dragons and had a bit of our power."  And then dragonborn were magicked into existence.

I'm thinking it simply come down to what players and GMs want to envision.

To me, boobs on reptilian and avian creatures would look out of place, regardless of any argument of them carrying some human genetic component. Same goes with bipedal fish, or insectoid folk. Protruding jiggly orbs of flesh with nipples would also look out of place on humanoid cetaceans, who's females most certainly would carry mammary glands. As a GM, in my worlds, if these folk exist at all, they are flat-chested.

Some folks feel differently about them. If a player in a world I'm running wants to envision their character as a busty dragonborne, or lizardfolk, or kobold, or . . . , I'm not going to worry about it unless and until they insist on describing those breasts, and their motions, during role play.

This a very thoughtful reply. I should have distinguished between human mammaries, and mammaries in general. I was talking about mammaries in general. Although, breasts from the human species are not just evolved sweat glands but also adaptations to bipedal movement and social grouping and interactions. One being the mimicing of the buttocks. Now that we stand up, face forward, and generally look at peoples faces when we look at them, cleavage right under their head is the next best thing. 

I will agree there is a reason, actually millions of reasons why a reptile would never form breasts. They all have to do with how they evolved before and are likely to evolve into. It's tricky when reptiles have essentially backed themselves into a proverbial corner. They chose the qauntity over quality approach, and their general lack of maternal bonds (save some species), theyvwould pretty much need to scrap their entire operating system. 

I don't want to make this too long, but my point being is, it is not impossible, and if life exists elsewhere, could actually be the norm or vice versa. Is it unlikely to see a reptile from our planet and ecosystem evolve a very specific gland on a specific body part in a specific way? Absolutely. Is it impossible? Absolutely not. Is it genetically impossible to imagine a snake with mammary glands all along its body, no but that would be strange and impressive.

Quote from SlimeMcGrime>>

This a very thoughtful reply. I should have distinguished between human mammaries, and mammaries in general. I was talking about mammaries in general.

Although, breasts from the human species are not just evolved sweat glands but also adaptations to bipedal movement and social grouping and interactions. One being the mimicing of the buttocks. Now that we stand up, face forward, and generally look at peoples faces when we look at them, cleavage right under their head is the next best thing. 

I will agree there is a reason, actually millions of reasons why a reptile would never form breasts. They all have to do with how they evolved before and are likely to evolve into. It's tricky when reptiles have essentially backed themselves into a proverbial corner. They chose the qauntity over quality approach, and their general lack of maternal bonds (save some species), theyvwould pretty much need to scrap their entire operating system. 

I don't want to make this too long, but my point being is, it is not impossible, and if life exists elsewhere, could actually be the norm or vice versa. Is it unlikely to see a reptile from our planet and ecosystem evolve a very specific gland on a specific body part in a specific way? Absolutely. Is it impossible? Absolutely not. Is it genetically impossible to imagine a snake with mammary glands all along its body, no but that would be strange and impressive.

I'm personally in the "no boobs" camp, but it's mainly because I prefer to emphasise diversity in the races – i.e- I don't like them to ever just feel like face-painted humans, so if a dragonborn is meant to be draconic/reptilian then I'd prefer them to be that way as much as possible while still being humanoid (a typically two armed, two legged, biped).

I guess it's more just personal preference, but I don't like race to ever feel like it's just a purely mechanical choice; that it can have impacts on how a character would approach and perceive the world, how others might treat them differently (more likely to view a kobold with suspicion than an elf, depending upon where you are) etc. A winged creature might find the behaviour of land-dwellers perplexing, or even difficult to fully understand, a tabaxi may wonder why everyone else isn't taking 18 short rests per day etc. 😉

I'm also a stickler for no tails, but that's partly because one of my dragonborn characters actually was a dragon trapped in a dragonborn-esque form by a curse, so him having a tail was unusual and I really wanted to emphasise that. Not that I'd stop a player from having a tail if they really wanted one, but I'd probably negotiate some reason for it, like being more akin to a half-dragon (e.g- more closely related to their ancestor), or maybe their lineage is unusual in having them or such. But as a rule my other dragonborn characters and NPCs are all tailless.

In my world, dragonborn do not have them because their ancient dragon creators do not idolize or fetishize humans or any other humanoid species for that matter. Dragonborn were initially created by benevolent dragons to be a less terrifying intermediary and/or emissary between dragons and comparatively tiny and delicate people. Malevolent dragons later saw an opportunity and followed suit for their own agenda. Breasts, frankly, would serve no necessary purpose in that role. If a player insisted on having them, I would be extremely wary and want to know why it is needed for them to enjoy the game.

For the most part, I also have no tails, though I do admit I did incorporate CR's tailed dragonborn for a very very small population who had served the dragons well and were rewarded with augmentation to be more like their 'kind of sort of gods'.