I would agree that there's an element of "I have an idea how this person will think about this" to "knowing" them, but I think that can be deceptive, because people are so fluid. I do think, though, that if a person consistently tells me (or indicates in some way to me) that I have been correct in some assumption about them, I start feeling as though I know them.

Of course, there's also the classic, "Oh, you know me so well!" from someone.