When people ask me where I am from, I say "New York" despite the fact they probably think that means I was raised in Manhattan. When people ask me what nationality I am, I say American. When they ask my ethnicity, I say Welsh American.
In truth, my father is Welsh, Irish, German, English, African American, Native American, and 100% Yankee. My biological mother was Irish, French Canadian, Italian, and Scottish. I was semi-adopted into my step-mother's family, and that is where the Welsh American comes in. My step-mother's mother was the child of Welsh immigrants and her father the child of Scottish immigrants. We lived in a community that still had strong Welsh cultural influences, and for some reason we sort of ignored my taid's ethnicity, and went with the Welsh bit. I had no real appreciation for how much that culture influenced how our family life went until I lived a while amongst the "non-Welsh".
I don't identify with all the other bits of my "heritage". I don't identify as multi-racial because people don't identify me as multi-racial. I am not ashamed of it, and I wouldn't have a problem if someone recognised it in me. I cannot say that I faced any of the challenges that my half-siblings whose fathers were black or latino faced, so I feel really weird even bringing it up. I was raised Welsh-American, and that's what I am. I don't really have a problem with people doing the "one bit this and two parts that" thing. I find it cute.
I have run into British people who bang on about their ethnicity or from whom they are descended (come on, I know many of you have met the great-great-great-great grand niece of Oliver Cromwell's best buddy or whatever) , so I don't think it's a uniquely American thing. Maybe it's the British influence (and the fact that Britain is such an amalgamation of cultures, both through immigration and conquest) that helped give the American tendency to do this to perhaps a bit more of an extreme sometimes.