During a recent campaign I run, one of the character uses their familiar constantly to scope out areas ahead. That's not a problem, and it's already well established that they can telepathically see through the familiar's eyes and hear what the familiar hears. The issue comes up when the character wants to apply their Darkvision feature to their familiar's eyes, which doesn't have Darkvision.
My understanding is that the character is receiving telepathically the familiar's senses. So if the bird hears a sound, the character hears a sound. If the bird doesn't hear a sound, the character doesn't hear the sound either. If the bird can see an object, the player sees the object. If the bird can't see an object (too dark to detect it), the player can't see the object either.

Do character features and traits apply to their familiar's via telepathic link? (For context: my player's are very adamant that they do and that anything scientifically reasonable never applies to D&D)

I do not think there is any official ruling on something like this, mainly because it is quite far-fetched.

Your interpretation is the correct one. This is a one-way senses-share, the spell specifically says "gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses."
That alone invalidates the claim of your player, as the spell clearly states that the master benefits from the familiar's special senses, and says nothing on the other way around, as well as specifying that the master senses are "switched off" during the senses-share. Usually, when something is not explicitly written in a spell, that spell cannot do it.

There is always the option of houseruling this, but as far as RAW and RAI are concerned, there is no share of special senses from the master to the familiar. (I do still feel bummed-out that they took away Darkvision for cats, though)

No, you're familiar has there own senses and abilities. This makes a familiar like the Owl particularly useful, as the player gains Darkvision by looking through the owls eyes. The converse is not true.

The spell text specifically reads: "... as an action, you can see through your familiar's eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses..." Your character gains the familiar's special senses and loses their own.

No you don't gain the advantages of your own statistics, e.g., Darkvision. More citations from the spell which very much hint at that:

Appearing in an unoccupied space within range, the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast.

Additionally, as an action, you can see through your familiar's eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses.

(bold by me)

For me this is as cut and dry as it gets. You use the statistics of the chosen form and are deaf and blind yourself, while seeing through the eyes of the form and hearing what it hears. You can not really write it much clearer. Your players just need to choose creatures with Darkvision if they want it.

That said, there's little harm in allowing it. Allowing the player to see through a hawks eyes with dark vision would mostly serve to broaden the players choices for a familiar. Otherwise he may feel the owl is the clear mechanical choice leaving him stuck with only one option.

Darkvision is a second level spell with a duration of 8 hours. I’d follow the RAW and make the PC cast Darkvision on their familiar if they want it to have Darkvision.