Well, yes the government does. Hence the term "dual citizenship." If the government didn't recognize the other citizenship, then there would be no such thing as "dual citizenship," as the person would simply be a citizen of the US and only the US, as far as US law is concerned.
My point was that dual citizenship = "natural born US citizen" for constitutional purposes, is not questioned by any member of Congress. IOW, those babies born in the US on the "citizenship vacations," then raised by Korean or Japanese parents, in Korea or Japan, for their entire childhood and young adult life, are (according to Congress) just as much qualified to be president (by dint of being born on US soil), as a child born to US citizen parents.
I believe this policy is new, is enabled by public ignorance, and is an incorrect construction of the US Constitution. Not that the Constitution is worth spit to Congress - just saying, this is another place where the federal government is way out of its bounds.
I believe this is exactly the US legal position. While other countries may recognize or grant citizenship to whomever they wish, we don't pay any attention to it. It doesn't change the person's position under US law.
A Supreme Court case way back in the 19th has some bearing. A young man born in this country to Chinese non-citizen parents (Chinese immigrants weren't eligible for naturalization at the time) sued to have his citizenship recognized.
One of the dissenting justices pointed out that if we recognized him as a citizen despite his ineligible for citizenship parents, we would be opening up the presidency to "coolies," thus pretty clearly indicating this particular man was "natural born" despite his parentage.
I don't believe the other justices specifically addressed this concern in their opinions, thus implying that it was indeed a possibility and they weren't too concerned about it.
BTW, I would love to see a random Hawaii birth certificate from the era, just to see what information it typically contains. Do you know where such is available?