Maybe he would have had to show a foreign passport to register as a foreign student (I doubt it, but it’s possible).
But that has nothing to do with his US citizenship, since the US allows dual nationality. My wife is eligible for an Irish passport, for example, through her grandmother.
If she went through the process and got it, it would not affect her US citizenship one bit.
And further, the US citizen parent of a child cannot purposefully renounce that child’s citizenship.
So, suppose Indonesia, or Lower Slobbovia, grants YOU their citizenship. The State Department’s position is “Fine, do whatever, but he’s OUR citizen until he says otherwise”, and that’s US law as well.