I don't think it's any different to , for example, South America.
I can only speak for myself, but I would refer to folk who live there as Argentinian, Brazilian, Mexican, and so on. Anybody from the United States I would refer to as American (Ethnicity noted). I would imagine that whilst living in America, you might refer to each other as Texan, or Californian etc, just as we would refer to each other at ground level as Southerners / Northerners, and more specifically as Mancs (Manchester) Cockneys (London) Scousers (Liverpool Brummies (Birmingham) etc etc. The nicknames would be used to discuss with others living here, as they would understand the terms, but in a more none familiar way to folk from other countries, we would use the specific areas.
It's all about categories, and the need to identify. If you look at an address, it will start with the smallest factor, such as house number, and expand via street name, town, City, Country, and so on. You, as an American, don't need to know where we live in such detail, so you will blanket any categorization, in inspecific format, such as calling us British, but it has a similar effect of shouting "Hey, You" and getting the reply "Don't say 'hey you', my name is Tom".
I would only call somebody European when talking about somebody in a none specific way, as in "They're not African, they're European" but I would respect them enough to be more specific, such as 'Spanish' or 'German' if I were referring to individuals, or nations.
If we are called 'Brits' then there is no recognition of our respective countries, and each of our countries have their own deep rooted histories, not to mention some even having their own language, so it is respectful that they should have their own identities acknowledged.