What does it mean when someone says something is their aesthetic?

What is the speaker telling the grecian urn in these lines from “ode on a grecian urn”? he wishes the urn could talk so he could hear its stories. he knows the urn will still be around for others to see after he is dead. he fears that when he gets old, the urn will suffer because nobody else will take care of it. he wishes the people frozen on the urn could warm up and come to life.