I was thinking along the lines of, well, it's not as if any given country "owns" somebody's name -- it's a property of the person, right?

That's a bad train of thought. You have to think about the institutions involved. There are certain things that international law guarantees to you, that your country is obliged to provide to you.

Things like "Everyone has the right to recognition everywhere as a person before the law." "Everyone is entitled in full equality to a fair and public hearing by an independent and impartial tribunal, in the determination of his rights and obligations and of any criminal charge against him."

In this case also important: "(1) Everyone has the right to a nationality. (2) No one shall be arbitrarily deprived of his nationality nor denied the right to change his nationality."

You don't really have an inane right for two nationalities. If a country allows you dual citizenship it's a nice thing to do. As such I wouldn't expect naming right to arise as a consequence.

As in, you can't have one legal name in one country and another in some other country.

That's certainly not the case.

If I remember right you can't have the same legal name in South Korea as in Germany or New York.

In South Korea your name needs to be written in Hangul and the legal documents about you are addressed to the name in Hangul. In Germany your name has to be in the standard Latin alphabet (I don't know how much accents it allows). Quick Googling suggests that the case for China is similar. You get to choose between Simplified characters or Traditional Chinese ones.