I grew up in Greece, NY, and I cannot answer that question. As I recall, we called ourselves "Rochesterians" because we were adjacent to Rochester. I can tell you that the local high schools are named Arcadia, Olympia and Athena. (No joke.) Other than that, Greece is just like the rest of upstate NY, that is, buried in snow 6 months of the year.
Matt, your town took on Greece and Greek names in the early 1800's to show their support for the Greek struggle during Greece's fight for independance from the Ottoman Empire. Just an FYI. ;)