In Tybalt's case, it means bloody control of the local Court of Cats. He became their king by right of blood; he's held the position by beating the crap out of anyone who tries to take it away. The Cait Sidhe take a more direct and bloody approach to succession than most of Faerie.

I’m a cat. We aren’t required to make sense.

When Rome burned, the emperor's cats still expected to be fed on time.

Cats never listen. They’re dependable that way; when Rome burned, the emperor’s cats still expected to be fed on time.

Most cats don't like water, and despite her pretensions of tigerhood, Julie was no different. Yes, tigers have stripes; so do tabbies. If you want to know the difference, try tossing one of each into your swimming pool. Then I would recommend running.