Per this article, we have a great sentence:
Matthew intentionally, intelligently, and persuasively avoids any discussion of “animal rights,” writing not about the putative “rights” of animals but about the moral obligations of humans.
I have said for a long time that you don't "own" your pets. Instead, you are "responsible" for your pets. Much as I love my dogs, I think it's fair to say that animals don't have rights, per se; we have duties toward them as living creatures capable of joy and suffering.
Similarly, I have said that there is no such thing as "State's Rights," only "State Powers" (as delegated to them by the People).
I find these kinds of distinctions useful, not for pedantic reasons, but so that we can think more clearly about what actions are appropriate and effective.