Americans, as always, do everything in bizarre ways. When American commies say "race is a social construct", everyone laughs, but they actually right, really. "Race" in their - American - terms is purely a social construct that has weak relation to ethnic/cultural/genetic concepts at all. They just use that term as default and don't understand other meanings (at least average generic person, of course more educated people exist in USA too).
Obama is not only half-black, he is also very distantly related to most of USA blacks at all. Most of "African-Americans" are descendant of west Africans - but Kenya is east Africa, populated by pretty different people with different history, culture and even genetics. He related to common American black person as Scandinavian may be related to Afghan - they both have white skin and may have blonde hair and blue eyes, and nothing more - everything is different from culture to ancestry.
So, he was pretty foreign to average black as any other USA president. He was ever more foreign, because his father is pretty recent immigrant. Election campaign was based purely on his skin color, but this was enough for so-called minority voters to think that he is related to them, and for average US white voters, who judge people by skin color as always. Cheap and fun trick, but it worked.
And there are plenty of fun things in US "race" classification. Arabs considered as whites for statistical purposes, for example, but Hispanics are separate category. Although it is very fun to see when some Argentinian with German ancestry considered as specific minority, but local US Germans are truly "white". And immigrants who came directly from Spain aren't Hispanics too. Hispanic term was invented long ago just for US-Mexican war purposes, they now try to separate "white Hispanics" and others, but it still a mess.>My point is, that he (and others in similar shoe) treated as if they weren't anything else, but black.
It is a cultural thing and specific media trick for voters. In some other country, like South Africa, he would be treated as "coloured" and general black population may view this as a bad thing. But we are sitting in American cultural sphere, and he is "black".
It is better to stop caring about this at all just because it is some kind of cultural circus that has weak relation to reality.