They are indeed similar in some ways; as you have noted, the kingside structure that gives the variations their names are identical to each other, and there are some similar themes, such as a kingside attack.
A comparison between them provides a great example of the general difference between White openings and Black openings. Black in the King's Indian Defense has to be very precise and constantly runs the risk of falling one move behind White with potentially disastrous consequences. On the other hand, because White started with the advantage of the first move, when he plays an initially restrained setup like the King's Indian Attack, all he is doing is ceding some of that first-move advantage, and the game does not balance on a knife-edge as much (until maybe later!).
For this reason the two openings proceed along quite different paths, and expertise in one does not necessarily translate to the other.
(To answer the nomenclature question: Indian in chess opening terminology tends to indicate a fianchettoed bishop (one on g2, g7, b2, or b7); the King's part means that the fianchettoed bishop is the one on the kingside; and Black openings tend to be Defenses while White openings can be Attacks.)