>But logically, there's no reason to believe that other animals can't be possessed by spirits.

Logic has no merit when you're talking about the supernatural, because the rules can be whatever you want. Maybe animals can't be possessed by spirits because God only gave "true souls" to humans, and possessing spirits can only replace "true souls." Maybe they require a brain with a certain complexity which only humans possess. Maybe spirits just think it's gross, like bestiality.