I think there's a definite difference between pet play, where the person roleplays the animal' and the having a human pet.

When I have a pet, I don't make her "sit" or "roll-over" or "heel"
I don't make her eat from a bowl, or sleep in a cage, or bark/meow.

But she does get trained. She gets to wear a collar and leash and may be required to crawl... She does take positions. And she must show her obedience. It's not animal play... but she is my pet.