ties that bind
Not much is known about Kairi's family. The one family member we know about is her Grandmother, as seen in a flashback in the first Kingdom Hearts game. This is when Sora somehow sees this, hears the story about children and their heart full of light, and then refers to it later with Kairi. None of the trio seem to mention their family at all. She may have been with her parents in Radient Garden, but perhaps they may have been part of Ansem's work, since she was made a Princess of Heart. That may be why she is seen with her grandmother, and remembers her.I personally think Kairi would have a family of sorts on Destiny Island, perhaps an adopted one or something. This is complete guesswork but if she was in an orphanage, she would have mentioned it, and she seems too well cared for to be someone who grew up in an orphanage. I could be wrong, however.