It sort of is, although perhaps more accurately you could call it a "coming-of-middle-age" story. At the beginning of the story Gilgamesh is an adult, but thinks like an immature wunderkind, just wanting to go off on awesome adventures with his friend to gain fame and glory. Then, once Enkidu dies, he goes on an existential tangent about the meaninglessness of life which motivates him for the rest of the story. It's after his failure to gain immortality that he is able to return to Uruk and resume kingship, understanding that true glory in the human condition is found in the propagation and continuity of culture and civilization, symbolized by the walls of Uruk that will endure after his death.

Or that's one theory.