Obama was born under the allegiance of his father's nation, which made him a citizen of that nation, he must be naturalized to be come a citizen of the US.
Of course the Constitution does not recognize a dual-citizenship status. If it does please reference.
“[I]t has consistently been held judicially that one born in the United States and subject to its jurisdiction is, from birth, a citizen of the United States; that such citizenship does not depend upon like citizenship of his or her parents, or of either of them (except in the case of the children of ambassadors etc.).
United States v. Richmond, 274 F. Supp. 43, 56 (CD Ca 1967). See also
Von Schwerdtner v. Piper, 23 F. 2d 862 (D. MD 1928)(child born in the United States to German nationals)
A person who is born in the United States, regardless of the citizenship of his parents, becomes an American citizen not by gift of Congress but by force of the Constitution. U.S.C.A., Constitutional Amendment 14, Section 1. In re Gogal, 75 F. Supp. 268, 271 (WD Pa 1947)
As such, the allegiance of parents whatever their situation is irrelevant in determining the citizenship status of a child born in the United States.“ At common law, a native is a person born within the jurisdiction and allegiance of a country, irrespective of the allegiance of his parents, except the child of an ambassador. Ex parte Palo, 3 F. 2d 44, 45 (W.D. Wa 1925)
(internal citation omitted)
Moreover, The Naturalization Acts did not apply to children born in the United States of aliens parents for the simple reason that they only applied to foreign born persons.
"this power granted to Congress to establish an uniform rule of naturalization is, by the well-understood meaning of the word, confined to persons born in a foreign country, under a foreign Government.", Dred Scott v. Sandford, 60 US 393, 417 (1857).
Applying the rationale in the Dred Scott case, children born in the United States to alien parents were not born in a foreign country under a foreign government.