radiantly
It is revealed late in the first game Kairi's true residence, which just happens to be Hollow Bastion. Sora sees a vision of Kairi when she was a small girl, listening to the story that her Gradmother tells her (which also features on the first pages of the manga, about children and light to the worlds). At this point, we, the viewers, have no idea where Kairi is. However, after coming to your final destination, you recognise the library as the one Kairi was in.It appears after these events Kairi regains her lost memory of her childhood, as when she is revived from her sleep she remembers the story and her Grandmother. Now, why did Kairi get shipped off to Destiny Islands in the first place? Because she was a Princess of Heart. Xenahort decided to use one of the residants as an experiment to 'get close to' or find the keyblade, which, she did. She met both boys who had a chance to inherit the keyblade.
She never question's her past life in Radiant Garden during the second game. She believes for herself that Destiny Islands is her true home, where all her memories lay. It is quite likely that Kairi lived in Hollow Bastion when it had its previous name, Radiant Garden. She probably lived closely to the Final Fantasy characters inhabiting the vacinity; that is, Yuffie, Squall, Aerith, Cloud, and possibly others.