Hah - well you made a point with your links that I didn't want to bring up, but since you did: the only people calling themselves "European" are people who really detest the idea of Brexit. But let me ask you something, what possible question could you be asked about yourself that would result in the answer "European". Here let me try:

Where are you from? > "Europe" is too vague, people want country or city.

What would your cultural heritage be? > Again, "European" is far to vague. Mine's English and Irish.

I'm really struggling here...

Again, if I didn't make myself clear: the use of the term "European" is entirely political (pro-EU).

As well as some 48% of voters in the Brexit vote. They are talking nonsense. Furthermore however much the Brexit fanatics may wish otherwise, the British Isles are still part of the continental landmass of Europe.