My belief is that Mama Dunham got her own mother's help to establish baby Barak's birth certificate and social security card (believed to be a fraud, too, but that doesn't matter either for this discussion) and thus presented a plausible set of paperwork for US citizenship at the time. Nobody really cared until he started running for office.
*correction: Not ‘abroad’, but “either abroad of simply too young to accumulate the required years”