I know that all men and mer were Ehnlofey and that the Argonians are Hist, but that makes the Khajit very strange. I know that Azura was involved but numerous deities were involved in the shaping of races and even orcs are still mer looking.

They also don't exactly fit very well into the whole Lorkhan vs Auriel war in Tamriel

They are, like men an mer, descended from the Ehnlofey. Probably.

You don't find it interesting how important the moons are to them? They don't, perhaps, fit neatly into said war, but then not much really does. According to their own myths, they come from the same "branch" of the tree as the elves, so take that how you will.

They're ehlnofey. More specifically from the same branch that the Bosmer come from. Their creation myth says Azura transformed them from "the forest dwellers", and the Bosmer pact with Y'ffre is a response to this daedric meddling.