The history of ancient civilizations in Mesopotamia has been a complex one, with several empires spanning a couple of millennia. Were the people from latter empires, say Neo-Assyrian or Achaemenid Empires aware of the existence of earlier empires, say Akkadian or Babylonian Empires?
The Babylonians and Assyrians had several versions of a king list, at least one of which enumerated the kings from the Old Babylonian period down to the Neo-Assyrian period. There is also a much older Sumerian king list, copies of which were discovered in Neo-Assyrian sites, so it is evident that these texts were still being copied and read many centuries after the kings listed in them. So yes, the Neo-Assyrian and late Babylonian literati were very well aware of ancient kingdoms.