The cat comparison for Tybalt isn't that much of a stretch. Mercutio even recognizes it, and he nicknames Tybalt the "Prince of Cats." Mercutio knows that Tybalt is vain, violent, and quick tempered. He sounds a lot like all of the cats I have had experience with.
Everything about Tybalt screams cat and cat-like. He might appear calm on the outside, but it is barely contained violence waiting to spring. Think about Act 1, Scene 5. Romeo has crashed the Capulet ball, and Tybalt is walking around in a storm. He's ready to pounce on Romeo at the first chance. The only thing that holds him back is Capulet himself.
A dog can be told to heel, and it goes back to acting dumb and happy. A cat on the other hand always seems to hold a grudge and wait for an opportune moment to get revenge. Tybalt is the same way. He honors Capulet's wishes, but Tybalt walks away with the full intention of slyly getting revenge against Romeo.
"I will withdraw, but this intrusion shall,
Now seeming sweet, convert to bitt'rest gall"