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.
They were well aware of earlier empires, and many Kings of the subsequent empires would intentionally style themselves as the 'rightful successors' to the previous ones, by taking the names of earlier monarchs(eg. Sargon), or using many of the same titles(eg. King of the Four Quarters).