I have an Aarakocra Monk character that adds it's strength modifier to it's unarmed attack damages, but the STR saving throw is marked as having proficiency (I don't know why... ?)
and therefore the damage is +STR +Proficiency... is that right or is that a bug?

eg. STR +1, Proficiency +2, Proficient STR Saving Throws = +3
= Talons 1d4+3 slashing

Yes you have proficiency with STR saves. No, you will almost never add your proficiency to damage. However, Monks have the ability to use their DEX instead of their STR for unarmed strikes. If your dex is +3 then that would make sense. 

If that didn't solve the issue;
Can you link us to your character sheet?
Are you using any homebrew?
Have you augmented the attack in someway?

Please post a link to your character sheet so someone can look into it for you, preferably the actual URL, not the “share link.”

Monks are proficient with Str saves and Dex saves because Monk. See:

Proficiencies

Armor: None
Weapons: Simple weapons, shortswords
Tools: Choose one type of artisan’s tools or one musical instrument
Saving Throws: Strength, Dexterity
Skills: Choose two from Acrobatics, Athletics, History, Insight, Religion, and Stealth

 (https://www.dndbeyond.com/classes/monk#ClassFeatures)

Monks can also use Dex instead of Str for their Unarmed Strikes because of Martial Arts. See:

Martial Arts

At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are shortswords and any simple melee weapons that don’t have the two-handed or heavy property.

You gain the following benefits while you are unarmed or wielding only monk weapons and you aren’t wearing armor or wielding a shield:

• You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.
• You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.
• When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action. For example, if you take the Attack action and attack with a quarterstaff, you can also make an unarmed strike as a bonus action, assuming you haven’t already taken a bonus action this turn.

Certain monasteries use specialized forms of the monk weapons. For example, you might use a club that is two lengths of wood connected by a short chain (called a nunchaku) or a sickle with a shorter, straighter blade (called a kama). Whatever name you use for a monk weapon, you can use the game statistics provided for the weapon in the Weapons section.

(https://www.dndbeyond.com/classes/monk#MartialArts-227)

So, your character is most likely using Dex instead of Str because, according to the Aaracokra, they can use their Natural Weapons as Unarmed Strikes. See:

Talons

Your talons are natural weapons, which you can use to make unarmed strikes. If you hit with them, you deal slashing damage equal to 1d4 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.

(https://www.dndbeyond.com/races/aarakocra#AarakocraTraits)

I’ma go out on a limb and guess your Dexterity score is either 16 or 17; am I right?

Monks have proficiency in STR saving throws. However, that shouldn't be relevant to attacks. Talons (the aarakocra feature) specifically do 1d4+STR, but this is an unarmed strike, which as a monk you can use DEX instead, and the DDB character sheet will pick whichever one is higher. So I'd assume you have +3 DEX.

Thanks very much guys!
So it's adding the DEX saving throw modifier for the +5 hit bonus and the standard DEX modifier for the +3 damage... ?

https://www.dndbeyond.com/profile/HarmlessBystander/characters/51495746

Saving throws have nothing to do with the damage you deal. Full stop. The reason that your Saving throw is the same as your to-hit modifier with the attack is that they are calculated the same way. Stat+proficeny bonus if proficant. There are things that will change the to hit without changing the saving throw and vice versa.

To-hit it will add your proficiency bonus, the damage will not. I highly recommend that you read through the Free Basic Rules here on DnDbeyond. They do a decent job of explaining a lot of the basics regarding the game such as character generation and class abilites. 

Saving throws are not involved in any physical attack you make to do damage. Saving throws are irrelevant to any part of the calculations you're asking about here.

1. Figure out if you're proficient (yes, because natural weapon). 2. Figure out if monk martial arts applies (yes, because unarmed strike). 3. Figure out whether you want to apply DEX instead of STR (which 2 allows you to do) (Yes, because 3>1)

Attack bonus calculation: DEX+PROF = 3+2 = 5

(If you weren't a monk, and therefore had to use STR, then it'd be STR+PROF = 1+2 = 3)

Damage calculation: weapon die + DEX = 1d4+3

(If you weren't a monk, and therefore had to use STR, then it'd be weapon die + STR = 1d4+1)

Quote from HarmlessBystander >>

Thanks very much guys!
So it's adding the DEX saving throw modifier for the +5 hit bonus and the standard DEX modifier for the +3 damage... ?

https://www.dndbeyond.com/profile/HarmlessBystander/characters/51495746

 
No, it’s not using your Dex Save to determine your attack. Your attack modifier is Dex modifier plus Proficiency bonus. Since you are proficient in Dex saves, that is also being calculated by adding your Dex mod + PB. 

Anything you are proficient in/with adds your Proficiency bonus. Anything that requires Dexterity uses your Dex modifier. If you happen to be proficient in Acrobatics, Stealth, or Sleight of Hand, they will use the same two numbers and also be at +5.

Here, read this: (https://www.dndbeyond.com/sources/basic-rules) Focus on chapters 7 & 9, they’re the most used and will explain pretty much all of this.

I hope that helps.

Thanks guys I'd already read those chapters, that's why I was asking because I knew it was wrong that saving throws should have anything to do with either.

What threw me is that the proficiency was still added to the hit modifier (thanks PocketMouse) because presumably unarmed-strikes/"natural-weapons", fall under the "Simple Weapons" category?

No, you have proficiency with unarmed strikes:

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.

Thanks Naruhoodie. I'd missed that. Makes way more sense!

A little strange even high elves and wizards are automatically proficient in unarmed strikes.
I'd been combing through the Monk and Aarakocra traits checking for that.

Well, everybody gets regular practice making contact with targets using our hands (picking things up, grabbing an item off a shelf), but not everyone has practice making contact with targets using a weapon. Think of it as the difference between catching a baseball and hitting the ball with a bat.