That's the player's problem to solve. I was simply stating that I would allow them to try, as long as they understand that the animal may never again advance as a familiar once it chooses to advance using XP and class levels.
The fact that an owl, by virtue of its anatomy would make a crappy wizard is not the GM's problem to fix. If the player wants a crappy wizard following them, that's their call.
In any case I used owl as a random example. An awakened Monkey should be able to do everything a normal wizard does, except look dignified and intimidating. :)