I think it's fair to say that he self-identifies that way in large part because that's how everyone identified him growing up. Hell, they still do.
But over the course of his career he chose to go along with that identification. I won't deny that he'd look stupid if he kept saying, over and over, "I'm not black, I'm half-white," because it's obvious that his phenotype-- that is, what a stranger can observe about his physical appearance-- has been taken from his father. To insist on your whiteness when you have dominantly black physical characteristics is rather pointless.
The thing you're attempting, though, is to critique American society because people don't automatically say, "He's not black, he's biracial" or some such noise. That might be fair if, once he ascended to fame, he explicitly defined himself as biracial. But he didn't do in his pre-fame autobio-- back when he was only famous for being the first black editor of the Harvard Law Review (or something like that)-- and he didn't do it as president, when he stated that Trayvon Martin could have been either "his son" or "him, 35 years ago." I don't even fault him for choosing to emphasize his black heritage. But by the same token I don't fault the average joe from saying that Obama was the first black president-- because in terms of his dominant phenotype, he was.
A contrasting example would be Alexandre Dumas. He had black heritage, but it didn't appreciably show, so his phenotype was "white." I've seen some allegations that he may have encountered prejudice for having been of mixed race, but that's not enough for anyone to assert that he was a "black writer."