A good question. Do pets have rights besides what their owners think? Certainly the owner can be arrested for animal cruelty, and that implies rights of the animal separate from their owners. In Washington, DC, dogs kept outside have to have a certain amount of space to walk around in. And your dog can be seized by DC Animal Control if it is left outside in weather that is too cold.
So your pet has certain rights no matter what the owner thinks or does. So in that sense, the question has already been answered. Yes, it would be incorrect to consider pets strictly personal property.