how do I take a pet off of favorites? like when you click the heart on their page

One pet must be active, you can't switch it off, only change which one is active.

you cant, they aren't saved anywhere in your account, there isnt a list or anything like that

I think they meant a "pet like" in someone else's pet profile ^^

Yeah, I wasn't sure. There's also pet friends, they may be confusing hitting the "like" heart and the list of pet friends.