Two of the dogs in my game are definitely obsessed with toilets. Those are the only major obsessions that I notice. Phobias seem to start up after vet visits. Maaike Haas's roaming black cat became phobic of the shower among multiple other things. She's basically always afraid now after the preventative shot. She's technically a Bombay Cat but I say that she is a random domestic black cat.
Siobhan Fyres' family now has a Chesapeake Bay Retriever. She doesn't really have phobias but she'll need to go to the vet soon when I play her again so she will probably develop some. Siobhan's daughter grew up and moved to BB where she and her new husband adopted a Weimaraner puppy who is now an adult. I sent the Weimaraner to the vet for just a spay and avoided the shot. The dog is still afraid after the vet visit. I'll see how long that lasts. If it keeps happening then I may just go get her the shot as well.
I've gotten around to Joaquin Le Chien who has finally married, sold his party pad and moved to Hound's Head - the 64 by 64 mansion lot. I've said that his new wife comes from money and financial connections. He also made money from selling his former home which was in a great location. Anyway, he adopted an adult Cavalier who is already neutered named Henri. Hopefully Henri won't need to go to the vet for a while. He is currently not phobic. I'll admit that watching Henri kind of makes me want a Cavalier again. (I had one as a little kid). I'm not sure I'll act on that impulse but I'll admit that I'm as excited to visit that household for the dog as for the human sims if not more so.
I haven't really dealt with pet fears very much other than locking the cat out of the bathroom where she hates the shower. In my own life, I've been pretty lucky to not deal with many pets (sometimes I pet-sit other people's pets) with any significant fears.