I think that the way the question is written make it hard to answer because its all a matter of point of view, and of translation. But the answer is surely to be found with the invention of writing.

The best I could do is to take an exemple that I know from a course of egyptian archaeology, the early dynastic period. Narmer is considered as the first Pharaoh to have unified the two kingdoms of upper and lower egypt. But the two entities that existed before were monarchicaly ruled, and if there was some diplomatic communication between the two powers, they centainly considered themselves as equal (we have no written sources, but the titulature of Pharaoes that emerged from the unification (king of upper and lower egypt) is a strong evidence. Moreover there was no mention of tribal links in the Pharaoes titulature, but it's maybe also cultural. For the ancient Semits, tribe and ancestry is very important (and is still more important in the middle east than in the west) so if people were prone to link themselves to their tribe, the kingship that would result from it would be tribal. Even if the if the book of kings in the old testament is a reconstruction made centuries after the events, tribal references are everywhere.

Sadly I'm not well versed in Sumerian litterature to extent this answer to Sumer and mesopotamia in general.

Probably not; the early portions of the Sumerian king list are universally considered mythological based on the lack of corroborating inscriptions and the absurdly long regnal lengths. The first king on the list whose existence is independently confirmed by epigraphic evidence is Enmebargesi. Of course, it is necessary to not that the Sumerian King List is itself a very late creation which attempts to condense several independent dynasties and city-states into an orderly "succession of kingship".