The "Prince of Cats" isn't a character in a fairy story. It's a nickname that Mercutio and Benvolio - and some of the other Montague boys, I think - have for Tybalt. Tybalt is the "Prince of Cats", and Mercutio's line means, in fact
Don't think he's just the Prince of Cats! Oh no - he's way more than that...
Why does Tybalt have the name "Prince of Cats"? It seems to be because he is such a fast swordsman, because he keeps taking deadly risks, and because he has nine lives (as Mercutio says in his death scene). Tybalt, Mercutio says,
He fights as you sing pricksong, keeps time, distance, and proportion; rests me his minim rest, one, two, and the third in your bosom! the very butcher of a silk button, a duellist, a duellist!
And, when Mercutio has been hurt by Tybalt's sword, referring to his role as a Cat (Cat-u-let, maybe?) complains he has received " a scratch, a scratch."
In that death scene, moreover, Mercutio keeps teasing Tybalt using that nickname - a dangerous comedy which heightens the scene. So Tybalt is a "ratcatcher", and, when Tybalt asks what Mercutio wants, Mercutio replies
Good King of Cats, nothing but one of your nine lives.
Hope that helps!
"Prince of cats" is in fact a name from a fairy tale. It is from "Raynard The Fox" Search up on that and you will get your answer.
"Prince of Cats" means a good fighter, because cats are quick on their feet, just like good fighters are. Mercutio is a good fighter. (like, with a sword)