More compassionate judges now are rewarding sole Custody or joint custody based on the needs of a pet and the ability of an individual to take care of that pet. In order for cats and dogs to be considered non-property, they must be considered Living, Sentient, Thinking Beings. Unfortunately, there’s no hope for that in our lifetime as the most intelligent evolved species, Bonobo’s (Resemble Chimps) would probably be the first. But do you really want to go to court, spend money hiring a lawyer and tie up the legal system over ownership of a pet?
If you’re going to be wed, then set up a Separation agreement that clearly indicates who gets the pet in the event of a breakup. If you’re just dating, I would highly recommend putting something in writing about who is responsible for the pet and who owns it and get it notarized.