I don't know anything of the original epic of Gilgamesh story. But in Fate I've heard Babylonia. Sumeria, Urik, and Mesopotamia. I have no clue what it's supposed to be.
From what I know, although I could be wrong, he was a Sumerian King ruling the city of Uruk. Babylonia came after the Sumerians, and Mesopotamia is kind of a general name for the area.