I've never thought of there being a story behind Omega. TBH I think he is the random superboss of the game.
Maybe it was made along with Ultima Weapon in the Underwater Research facility, and upon hearing that it is the strongest monster ever, decided to transport it to her time and have it in her castle. I don't know, that probably seems the most likely story to me.
But like I said, I don't think there is meant to be a story behind what he is or why he is there. He is an allusion to the other games (which FFVIII does not have too many of), so perhaps that is all he is.
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