I mean, New Zealanders are less likely to use New Zealander as their denonym. Speaking from experience, you're more likely to hear Pakeha, Kiwi, or NZ European.
I've heard it a little in England, usually with the prefix Anglo- , such as Anglo-American, or Anglo-Indian.
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Not very often though, and pretty much only when talking with family.
As others have said- The United States is a country that was recently made of immigrants. But there is more to it. In Asia, within China people aren’t just “Chinese.” There are those from Taiwan, Hong Kong, Malaysia, the Cantonese, the Hakka, etc etc. China is big and knowing someone is from a certain part of certain city helps understand their culture and what dialect/language they speak, can help understand their politics etc.
The EU is relatively new, it’s maybe not that strong a union. The USA is kinda like an EU that has lasted longer and took the whole “union” thing very seriously. So it isn’t common in Europe for a person to say “I’m an EU German...” for several reasons. Chiefly is that for most Europeans the EU isn’t part of their identity, or isn’t strongly. They’re just “German.” The EU doesn’t really have a culture or identity yet. It might not ever.
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Not very often though, and pretty much only when talking with family.