Kahani is a place, not a person, in this story. Haroun finds out from Butt, the Hoopoe, that the celestial body that looks like "a large asteroid," is Kahani, the earth's second moon (page 67). Kahani moves so quickly that no instrument on earth has ever been able to pick it up. Kahani is covered with liquid, and shines with beautifully bright colors. 

Later, Butt tells Haroun that Eggheads in the Land of Gup figured out how to control Kahani's rotation. Therefore, the Land of Gup dwells in "Endless Sunshine," while Chup lives in perpetual darkness (page 80). Haroun realizes that this whole tale doesn't quite make sense, as if Kahani rotates around the earth, it must at some points be blocked by the earth between Kahani and the sun. However, Butt says to ask the Walrus about these types of questions. There is an element of a fantasy in the story about Kahani. 

Kahani is the inhabited second moon of the Earth.

Kahani is a moon that has yet been undiscovered by humans.  Haroun goes looking for more stories, and finds the Sea of Stories

All my life I’ve heard about the wonderful Sea of Stories, and Water Genies, and everything; but I started believing only when I saw Iff in bathroom the other night. (ch 8, p. 137)

The Sea of Stories is a beautiful place, full of colors.  The Water-Genie shows him the different stories, and how they are constantly changing.  He is given a wish-drink so that he can take the stories back home to his father.  Once he arrives in Gup City on Kahani, he sees that his father has gotten there ahead of him.  He also finds Khattam-Shud, who wants to control the world.