There isn’t anything preventing native-born Americans from holding dual-citizenship status, should they want.
Now, certain foreign nations require, as a prerequisite to holding THEIR citizenship, that any immigrant renounce their former citizenship.
Btw, it says "No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution...". In 1765, the meaning of natural born was BOTH parents of the same nationality, actually the father determined nationality.
I can't believe that you would think it's Constitutional for the President to have dual citizenship.