Did ancient society’s, such as the Egyptians or Greeks, know that they were the early in the history of civilization?

More can be said if anyone wishes to add something, but in the meantime here are a few older answers you may enjoy:

/u/toldinstone addresses what the ancient Greeks and Romans thought they knew about ancient ruins in Did ancient civilizations have ancient civilizations?
Because mythology is also sometimes relevant, /u/UndercoverClassicist answers addresses When did the Greek myths "happen" from the perspective of the ancient Greeks? Did the stories about heroes like Jason and Perseus happen in their ancient past, or did they believe these things were ongoing around them in real time?
/u/Antiquarianism does an amazing dig through a whole bunch of ancient cultures' understandings of their historical pasts—including various Native American, African, and Mesopotamian groups, but they really do touch across a lot of the globe—which can be found in another thread on Did ancient civilizations have ancient civilizations? (each link goes to a separate discussion on that thread)