Inside the US, we don't refer to ourselves as "American", we seem to go by whatever the hell our descent was, but outside of the US we refer to ourselves as "American".
Is America the only country that does this?
Also, on another note: To all you who are not native English speakers: When you buy a videogame, do you play it in English or your native language?
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The idea that a person in America is "1/8th Irish" or "1/4th Russian" is a fallacy, and people rarely do that once they are old enough to understand how illogical it is to say such things. There is justification if your parents are native to another country, or even your grandparents, but beyond that, you are an American through and through. Personally, I've only experienced others mentioning what state they are from. Are you from the south? The north? East coast, west coast, so on and so forth.