The only immortal thing is change. I think what you refer to is the ability to identify and track that "constant change". I understand that. It's much easer to track something red (that empire's name, that leader's name) in a sea of something green.
Even now, there is very few things connecting a Civ game empire on Turn 1 to an empire at turn 200. Empire's leader, color and name is the only thing that is truly the same. I guess you could keep a roman legion unit around in modern era, just for novelty.
Well, I think we could go off on a rabbit trail discussing my own personal beliefs I believe to be Immortal but I will restrain myself.

Anyway, this article touches on what I am attempting to convey. (perhaps not very well)
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Introduction to the quest for immortality
The quest for immortality has been a fundamental part of human culture since ancient times. It has captivated the minds and imaginations of people across different civilizations, driving them to explore various methods and practices in the pursuit of eternal life. From alchemy and the search for the mythical elixir of life to religious rituals and ceremonies aimed at achieving immortality, humans have left no stone unturned in their quest for eternal existence. Ancient philosophers have also pondered the concept of immortality, contemplating the nature of the afterlife and exploring the possibility of reincarnation. The quest for immortality has had profound ethical implications as well, raising questions about the meaning of life, the value of mortality, and the consequences of eternal existence. Despite the elusiveness of immortality, the human fascination with eternal life endures, shaping our understanding of the past and influencing our hopes for the future.
The Quest for Immortality: Ancient Search for Eternal Life - History of Yesterday
Overview Introduction to the quest for immortality The quest for immortality has been a fundamental part of human culture since ancient times. It has captivated the minds and imaginations of people across different civilizations, driving them to explore various methods and practices in the...

Perhaps playing an immortal Civ in a computer game is a reflection of that?