Most familiars can't speak languages (Ravens, Parrots, and Thrushes can). They can communicate telepathically with their master, but not being able to talk to the rest of the party or NPCs that aren't the same type of animal would present some difficulties.
Normally, familiars don't gain levels (or hit dice) on their own. As their masters gain levels, they gain bonuses spelled out on the familiar page. Things like increased natural armor, intelligence, spell resistance, and the ability to deliver touch attacks for their masters spells.
If your player isn't gaining class levels, they will be severely overpowered by almost everything else, as they're really just an extension of another character and not their own class.
If you do let them have class levels, you could wind up with a Wizard whose familiar is another Wizard, which would be... interesting. The primary problem in that case is if you would stack the familiar's bonuses with normal class levels? On top of that, you've got a party member that's a house cat, which would present some unique challenges in acquiring gear.
It's not impossible, but it would require some thought.
To get what you want, it might be easier to throw out some of the familiar rules. You could do something like have two PC characters where some kind of accident happens, and one of them is permanently polymorphed into an animal and gets linked to the other character. Give them the relevant familiar special abilities (telepathic link, scry on familiar, etc), but throw out the familiar stat bonuses and treat the character as a PC in the relevant animal form for those things instead. Also let them retain speaking ability, unless the PC wants to lose it. Not being able to interact with most PCs or NPCs at all can get very restrictive.
That's going to get your players what they want without leaving one of them with significant weakness in terms of how they can interact.