I have kind of the reverse situation. If I were anymore white I'd be transparent. I'm from the States originally, but I speak Dutch with some degree of fluency and ...am very white.
In everyday interactions my roots don't often come up. But if I talk to someone long enough after 15-20 minutes something about my grammar or accent will tip people off to the fact I ain't from around here.
Mostly people just complement me for my Dutch and add that most Americans don't bother to learn it (this however is changing rapidly since we have had so many people from N America move here and immerse as quickly as they can the last few years.)
But once in a while they will ask some rather pointed questions about things like religion - assuming I'm a crazy xian - or racism - assuming because I'm white and from the States I must be a racist - etc.
My wife (100% Dutch, couldn't possibly be more Dutch) works with refugees, and I've helped on many occasions over the years. One of them has become a good friend of ours. And to the refugees it definitely matters where someone is from and it's the first thing they will ask one another. It is sometimes critical to segregate people from different places or cultures, for safety. I think because of this they don't mind when whitty asks them where they are from also. Most will want to talk about their home country.
But, these are refugees, not someone whose parents immigrated here three generations ago and just happen to have brown skin. Asking a non-white person where they are really from is Not Done. But once you get to know someone they will probably bring it up in conversation.
All that said, some Dutch people can be hella racist despite our reputation for tolerance.
The Polder Model strikes again!