One thing that is common in the US is talking about heritage. A lot of people will say they are German, because their family came from Germany to the US some time in the past 200+ years.It doesn't really. no, at least not among white europeans. Europeans of middle eastern, african or asian descent are probably more like to talk and think about those things, but that's probably linked to the fact, that in many european countries people with heritages from those regions have problems being recognized as Europeans even if they were born and raised in a europeans country.
But I feel like that probably doesn't happen in Europe.
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