Not to kill the mystery, but she remembered her name as Kairi from when she came to Destiny Islands; we know because when Sora witnesses the memory of Kairi being spoken to by her grandmother in Hollow Bastion, her grandmother's speech ends with "Kairi". Remember? Sora goes, "Kairi?!" and reaches for the little girl, but he can't...quite...reach her. And then he wakes up in Traverse Town again, with Donald and Goofy.

Anyway, Kairi never said she forgot everything -- she just doesn't remember much about where she came from. ;) I think this is more a case of being very young -- she was only five when she left Hollow Bastion -- than having amnesia...