My kitty Max doesn't think he is a dog. He simply thinks that everything a dog can do, he can do better.
Max knows how to sit and beg on command. It was hilarous when I was trying to teach Zeus how to sit. Max would walk up to see what's going on, and start doing the sit right along with Zeus. It was as if Max was saying to Zeus "Hmph. I can do that. See, stupid?" I swear that Max taught Zeus to sit, and not me.
Max also loves to fetch his toys after we throw them. It's his favorite game, he'll play all day long if we oblige him. We tried playing fetch with Zeus, but he doesn't like it. When Max saw us try, he'd run over and grab a catnip mouse and bring it back to me. And he'd give Zeus the most smug, superior look I'd ever seen a cat give (and that says a lot!).