US law does not forbid dual citizenship. It does not encourage it, but there is nothing illegal about being a citizen of both the US and Kenya, for example. But you are correct that whether or not someone is an American citizen is purely an issue of American law- your status as a citizen of another country is irrelevant in that determination.
The U.S. State Department acknowledges that U.S. citizens in fact hold passports from other countries, but U.S. policy and law would not permit anyone to officially acknowledge a U.S. citizen's citizenship of any other country.
That's not correct. The US is aware that I am both a citizen of the US and Canada- it comes up every time they swipe my passport when I enter the US (the two countries share immigration information, so Canadian immigration officials see the same info when they swipe either my Canadian or American passport- I typically carry both when I travel).
U.S. Customs treats you according to the passport you present; if you were to show your Canadian passport after standing in the U.S. nationals line, they would tell you to go stand in the other line.
The U.S. government acknowledges that other governments may consider U.S. citizens to be citizens of their country, but the U.S. government does not consider that consideration to have any force in its own policies. Someone who is considered a citizen by both the U.S. and Canada, for instance, has no right to consult with the Canadian consulate if he is arrested in the U.S., for instance, if his U.S. citizenship is known to police.
Obama's mother was 18 when he was born in 1961. So the only question is whether Obama was born in the US. He could clear that up immediately by producing his live birth certificate showing the attending physician, name of hospital, etc. like McCain has done. I am not a conspiracy nut, but I do find it odd why Obama is fighting the Berg and other suits in court when he can clear things up very easily. The question then becomes what is he hiding?