To the extent that your presupposition is valid, Americans identify by race rather than ethnicity because they live as a "white person" or a "black person" or a "hispanic person" etc. every day. Ethnicity for most Americans is less relevant to identity because we're an immigrant nation divorced from what it means to live daily as Irish, German, Russian, English, etc. Mostly it's fodder for small talk and a way to choose the team of your secondary allegiance for the Olympics and the World Cup (or primary allegiance for the type that pretends to be Canadian when traveling overseas.)
But, as I alluded to at the start, your presupposition isn't entirely valid. As should be clear from the minor sniping in the thread so far, most Americans have a regional identity that's nearly as important as a general American or race-based identity. This regional identity may mean for Americans what a national ethnicity means for a European, especially considering the size of America and its states and the size of Europe and its nations. As long as they're of comparable economic classes, a white Texan probably has more in common with a hispanic Texan than he does with a white Oregonian. There are also certain ethnicities in America for whom ethnicity commingles with race to the point that they're inseparable, notably Jews and Mexicans.