When you argue for sovereignty, it's those people's sovereignty that you argue for. It's those people's right to choose their state, their government.
Sure, but when we talk about a "people", well, notice the grammar, I said "a people", not merely "people". The context is different. When we talk about "a people", it's a reference to a nation, or a culture, not just a random selection of humans.

Point is, you don't decide what culture you belong to. I know you're the kind of guy who believes in freedom of choice, which is awesome, but that only goes so far. I mean, I can't choose to be black, can I? So how can I choose to be South Korean?

Of course, if I spent most of my life in South Korea, me being a white European shouldn't itself be a barrier to belonging in Korean culture.

idk if this Kazak Russian tennis player has any legit cultural link to Kazakhstan, and you're right that it's not me that gets to define what is an isn't a legit cultural link, but I can have an opinion. And if this lady's wikipedia page is accurate and that she chose Kazak nationality for purely economic reasons, then in my opinion that's a tenuous cultural link. Feel free to disagree all you like, but unless you're of the opinion that I can indeed choose to be black, you have a line somewhere too.