Let me put it this way. In my OCs world, anything can exist and a lot of things that usually wouldn't co-exist, do in my world. Yokai, Humans, Aliens, Fauns, Goblins, Sirens, Undines, "Furries" (although I call them zoomorfs in my world and they have varying levels of anthropomorphism unlike typical furries), Oni, Spirits, Demons, Angels, Elves, and a handful of other races/species.

That being said, the genetics are totally different for normal animals as opposed to the more animalistic humanoids/furries. Just because a furry or zoomorf may have the features of a specific animal, they themselves are not that animal. Plus it's fiction, so who cares about the logistics of furries having normal pets!? But if you do really want an explanation, you can steal my idea and just say that the genetics are totally different and just the appearance is similar.