I can only speak for myself, but when I'm speaking Greek to family, I use the Greek version of my name to keep on the same cognitive track. I'm bilingual as well, and I only think of myself by my English name, but the other name is true too. Perhaps I'm a bit sensitive on the topic, but there's this very strange need to wash away Krypton and say "he's a HUMAN/AMERICAN" as if he can't be while embracing his heritage.
Ironically, I claim very little of my heritage in my day-to-day life. It comes up on these boards 1000% more than in my offline life, but nevertheless, I do see a consistent track along DC's history in downplaying Clark's heritage to enforce the idea he's "one of us" and it plays into a xenophobic vein very much a part of the USA.