I love Wyatt, my dog. He is not "my dog" in the sense that he is my property, even if most people, society, the government or whatever institution assumes he is. I care for him. It's that simple.
I think about this a lot because one time in summer camp a counselor had a dog with him and I innocently asked if the dog was "his dog." He said: "Oh I get this all the time; no he is not my property, I just care for him."
I felt judged and hurt. I remembered that moment well because I thought about and still think about it to this day. One thing I realized about the language of "my dog" is that it's the same as "my partner" or "my husband," but in these instances no one, whether asking or stating, assumes that property is implied. So why must it mean that in the case of dogs or other pets?
Well surely society in general does think this, government thinks this, many institutions assume this, and they use the same language to mean property, but as shown with people, it doesn't have to mean that.
So for me, I don't think I own my dog. As I said, I care for him. When people ask me if he is my dog I remind myself of all of these ideas and say, yes, he's my dog, knowing what it means to me and knowing that whoever asks is innocent no matter what they mean.