I didn't have any cats at the time of transplant, so my doc recommended I didn't get any. Had I already had them, he would have said the same thing your's did... just don't change the litter box b/c of the risk of toxoplasmosis. I was also told not to clean my dog's ears and not to have animals in the bed with me. (although I am guilty of letting my poodle sleep with me <img src="i/expressions/face-icon-small-smile.gif" border="0"> )
Reptiles and birds can carry salmonella, so I don't think it's a good idea to get one of those as pets post-transplant. I would think that if you already had these animals though, then you should just avoid cleaning the cages or wear a mask when doing so. I cannot see telling a patient to get rid of pets they already have. That's traumatizing to the owner and to the pet.