“Doggy” for dog makes sense, as does “kitty” for cat, when you think about it. (“Kitty” is a simply corruption of the word “kitten.”) But where does “bunny” come from?
Actually, “bunny” is closer to the original word for the floppy-eared critters than “rabbit” is. Up until the 18th century, the common word used for adult members of Oryctolagus cuniculus was “coney,” which was pronounced “cunny.” Eventually, that word fell out of favor because of its similarity to a word that was becoming increasingly more commonly used as a vulgarity. So timid souls along the way chose a rhyming alternative, and “coney” became “bunny.”
“Rabbit” was originally only used to describe a young or baby hare, but eventually the meanings of the words got switched, and we now use “rabbit” as the catch-all word for Bugs and his friends.