Greek and Roman law stems in part from the Bible. These laws find the universe was designed by a designer for the sake of man. All that is in the universe is here for the benefit of humans. The Law was established for man's sake. A legal right must be recognized by the law as a legal right.
Humans can and do have legal rights.
Non-human animals never had any legal rights in Western Law. Not now, not ever. There is limited law in favor of horses and food animals with regard to slaughter protection. The laws do not extend to any noticeable degree beyond the farm. We cannot, under the law as it currently exists, tell another how to keep pets.