right of publicity (also known as "likeness rights.") This is despite the proclamation by the IP guru, Melville Nimmer, that owners of animals should have such rights (scroll to page 73).
What about Lassie? It's possible that famous animals may have a trademark claim to their name and/or likeness (and certainly their owners have tried for various property rights). But, so far, no court has extended right of publicity claims to animals.