I am a lot of things to my animals: guardian, advocate and owner.
I do own them, just like I own many other non-living things. But, even some non-living things need care and maintenance (like cars, horse trailers, etc), so to say that ownership over something is just without those aspects is not necessarily completely accurate.
You can absolutely own something that requires to maintenance, like a rock, but you can also own something that has needs, like a car.
I think of my housepets as pets and companions. Their existence is purely to keep me company and for my amusement (cats do some really funny stuff, so do dogs). I see my horse more as a partner and companion since when I go riding, I like to think that we’re working as a team. He is more than just a pet because he works - he doesn’t just meander around and eat and tolerate me fussing over him.
I call all of my animals my babies and refer to myself as their momma. They are my loved ones and just happen to be non-human animals.