In America, a nation of immigrants, people are often proud of where their family came from. This can trickle down a few generations. There is an increasing number of people that identify as "American" instead of where their family originally immigrated from, particularly in Appalachia and the South, so this could eventually change.
I understand the frustration from a European point of view, but I just wanted to provide some context to the phenomenon.
Even when that’s true—-and it often isn’t--they usually mean that they’re part of a specific American subculture.
Italian-Americans, for example, traditionally celebrate Christmas Eve with a “Feast of the Seven Fishes.” This dinner isn’t traditional in Italy, but it’s also not common in other American cultures either. An Irish-American family might have a turkey instead but they’re more likely to “observe” St. Patrick’s Day by eating soda bread and corned beef.
It's a weird mindset, and it's definitely tied in to the toxic streak of white supremacy that runs through all of American culture.
Namely: Why would it be called "white EUROPEAN" when basically none of the people it is applied to are European in any reasonable sense of the word? They may have great-great-great-grandparents who were European, but that does not make them European.
Americans do this all the time - they call themselves "german", "irish", or "italian", even when they are not born there, do not speak the language, have never even visited the country, and have zero exposure to the culture of that country.