Politically, can you even be just "Korean"? That's like living in North or South Dakota/Carolina and just being "Dakotan/Carolinian". Geographically, culturally, historical, etc, they're very similar, but politically are distinct entities.

If you are born in Vatican City are you Italian? (Note that Vatican City is considered its own sovereign nation).

.... you are

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Comparing national to a state level is a bit misleading.

You can visit Gumgansan(The Diamond Mountains) in North Korea. Also some people(mainly press and diplomats) can visit Pyongyang.

I don't know about North Korea, but I would like to visit Seoul.

I plan to visit Gumgansan at some point... Seoul well I live there.

Vatican City is an oddity. And not sure quite what that has to do with my point. My point being that, in common speech, you can be Korean, politically there is no place called Korea at this point in time that I know of, only North Korea and South Korea. Just like you can theoretically be from Argentina or something and be "American" because you live in South America, thus saying that you're American (this is at this point really stretching, since it's not common in speech to use it like that) is not politically defining. Saying your Korean does not define you politically in regards to a specific country.