Yes, there have been several studies on animal self-awareness, and there've been some surprises. Here's an article with a (non-exhaustive I'm sure) list:
10 Animals with Self-Awareness
The magpie was the biggest surprise for me:
But...self-awareness is not what I'm calling full self-awareness. Koko the gorilla still refers to herself in the third person, IOW as "Koko". But the real difference is they don't have the resources to develop the idea of the ubiquity and permanence of death. Even if some animals were capable of that, not having that knowledge prevents them from becoming fully self-aware. Even humans aren't able to grasp that concept until we're around 2-3 years old--part of the reason for the terrible twos.
I'm not saying some animals aren't capable, and dolphins may have it. If so, they, like us, would be capable of morality, which some AIs may eventually develop. If so, that entitles them to equal rights with us, something that won't go down easy for some. Meanwhile, we give them our most humane treatment.