Communication Issues
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.
Class Levels?
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.
House Rules
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.
RAW (Rules As Written), it looks like it according to the rules for familiars and looking at improved familiar, improved familiars do not give you a bonus to skills. Not that this is a terrible thing; some of those improved familiars have super nifty abilities! They just don't give you more skill points to a skill. (Some of them could just perform the skill for you!)
However, one could argue that the improved familiars should grant skill bonuses, as per the "approximating familiars" section in the first link. This is all subject to DM approval; I doubt many of them would allow it.
Best Answer
These feats are all from the Pathfinder Player Companion: Animal Archive splatbook, which has a general note at the top of its section of new feats with the wording you quote there. Unfortunately, there is potentially some ambiguity as to which feats precisely this refers to and whether or not they have been correctly copied onto d20pfsrd.com.
Of the feats introduced in that resource, some of them specifically require the taker to be a familiar, and some are feats that a familiar is probably qualified for but don't have "familiar" as an explicit prerequisite. It's not clear whether the latter feats are "feats meant for familiars" and can be switched in this way, as they've not been consistently copied on d20pfsrd.com with the special note attached, and at least one of the former made it onto the site without the note.
Excluding feats with prerequisites that no familiar can satisfy, the list of feats a familiar could conceivably take from this resource are:
In Pathfinder Society play at least, it had to be specifically ruled that familiar could swap in Extra Item Slot, as noted in the Additional Resources rules:
The entry does not preclude other use of the feat-swapping rule, so one is forced to draw the conclusion that PFS at least did not read Extra Item Slot as being a qualifying option normally - I expect this why the feat entry on the online SRD ended up with the note whereas other feats from the splat didn't.
Personally, I'd judge that the intention was probably that feats are only "for familiars" if they have "be a familiar" as a prerequisite, but most of the others from this splat aren't particularly powerful choices anyway so there's probably no harm in allowing them. Extra Item Slot, however, stands out as the most powerful option available here, as the ability to use an extra magical item allows for much more powerful effects than the feats can offer.