I know many blacks who were born under Jim Crow laws; does that mean they accepted those laws? What matters is does he claim to be a British citizen. THAT is the crux of the matter.
Don't let your dislike for the man (and I immensely dislike him personally and his policies) cloud your mind about law.
Barack Obama, on the other hand, admits and acknowledges that he was born under the British Nationality Act of 1948 and that it GOVERNED his citizenship status. He is assenting to the fact that it governed his citizenship status.