>While Tesla was ethnically Serbian, I'd consider him to be an American scientist since that's where he lived for most of his life, and did most of his work. Plus he was born in what is today Croatia. Honestly, though, does it really matter other than for fictional nationalistic bragging rights?

Yes. It matters if one keeps a tally of what one civilization or another did, like we do for ancient times, and the future will do for us for example.

It also matters if one wants to get an estimation which country, kind of educational system, laws, etc, produces more or less good outcomes (e.g. innovation) in this or that field.

And of course, if the person claimed he is so or that, and cherished that, we should respect them, and not rebrand them. For example there are examples of famous immigrants to US, France, and other places, in literary and other circles, that don't like to adopt their country of citizenship as their nationality. For them their culture is always their other one, and their citizenship is only a matter of residence, convenience etc.