It's not an issue I've run into in practice with Europeans, either, who mostly seem to use cognates of "Americans" in various languages. Greeks call Americans Αμερικανοί, the French call us américains, the Danes amerikanere, etc., and nobody seems to think twice about it. Even Québécois typically use américains as the demonym, so the fellow-North-Americans issue doesn't seem to be a big point of confusion. I've seen "USians" on the internet a lot (along with "UKians"), but never encountered it irl. I'm not sure how you'd even say something like that in Greek.
If anything, my experience is that the Europeans I've interacted with use it a bit more widely than even Americans do. While we universally use it to describe the people, many Europeans seem to use it as a name for the country, "America", in places where I personally find it weird. Outside contexts like national holidays and poetry, I usually say "the United States" or "U.S.", as in "I'm flying back to the U.S. next week", but many Europeans I know say things like, "this is my first trip to America" or "I have a brother in America".