Well, it's an easy enough conclusion to jump to. After all, Leon and Aerith tell Sora, that Ansem the Wise ruled Radiant Garden, and we see a flashback of Kairi and her grandmother in his library (on a completely random note I bet it was Ienzo who designed those maze book shelves).
And so AtW, being a seemingly benevolent ruler, wouldn't let peasants into his library if they sought knowledge? Or couldn't Kairi's grandmother have been a servant there or something along those lines?
And Xehanort's Heartless confirms that Kairi is a princess.
This has already been adressed, and I'll say it again:
Being a Princess of Heart does NOT make you an actual princess. It just means you have no darkness in your heart. That's it.
He said despite Alice and Belle became a princess after loving and marring beast who is a prince. Most disney princesses are not born princesses.
It's half and half in KH. Aurora, and Snow White, and Jasmine were born princesses. Belle, Alice, and Cinderella were not. We don't know about Kairi, but likely the answer is "no". And Belle isn't even a princess in KH until the end of KH2, so that point is irrelevant anyway.
And if you read a book series called the Looking Glass Wars Alice is a princess.
Which has nothing to do with Disney's adaptation of Alice in Wonderland, making this not matter.