First of all, it's Kairi, not Kari. I don't know if that was just a typo, but figured I'd point it out.
I don't really remember her backstory, though. It's been too long since I last played any of the games.
As far as the Disney Princesses go... There are some who are actual princesses, princesses by marriage, some who are "honorary" princesses, and some who are honorary Disney. Examples of each:
Princesses: Aurora, Jasmine, Rapunzel, Pocahontas, Ariel, and Snow White
Marriage: Cinderella, Tiana, Belle
"Honorary": Meg, Mulan--this is more because of the part they play and their characters than anything else
Honorary Disney: Anastasia