I’m not sure this is correct; in general many people would think of themselves as X and European, where X is the country of origin or citizenship. The EU is not a country.
For that matter, the UK, despite being a country, has some aspects of this; while Northern Ireland in particular is complicated for historical reasons, plenty of English and Scottish and Welsh people would think of themselves as English or Scottish or Welsh AND British.
For that matter, the UK, despite being a country, has some aspects of this; while Northern Ireland in particular is complicated for historical reasons, plenty of English and Scottish and Welsh people would think of themselves as English or Scottish or Welsh AND British.