How would you describe the political organization of the city of uruk?

Gilgamesh was the supreme ruler of the city, a part-god, part-man. The best modern political designation would be close to an absolute monarchy. Meaning that there is one rule whose power is absolute without a system of a constitution or other laws. The decisions made by the leader, in this case, Gilgamesh would be absolute, even if they were horrible or crazy.