right now i have 2 main theories.
he was always Kreis; the book he wrote was a half remembered vision of his past life. (this is why professor keeps telling him not to worry about being a doubleganger). alternatively, we also saw in this chapter that stronger feelings can send him further back in time, so maybe his "current" self was kreis' past life, and him thinking up the story was the point he jumped back to.
i don't think that kreis' revenge is the "wish" he's there to fulfil based on the fact that siegfried becomes an apocalyptical, world destroying monster when he looses kries, i think the wish he's there to fufill is a wish seig may have had at the end of the original timeline. his wish is, presumably, the sentiment that that things were different. e.g "in another life, i wish i could have been happy with the one i loved" or "i wish i could have spared him from this fate"