Yes I feel European.
Every part of my life is a mix of elements from different European cultures, with some non European thrown in but on a day to day obviously level more European.
I live in Germany and most days only speak English to my family except when teaching it, German to everyone outside the house - I have two jobs, for one I only speak German and local dialect (the other is teaching English obviously).
My husband is German - ironically he spends his entire work day speaking English because it is the business language in his company. He reports to an Irish boss and has colleagues who are mostly from Italy, Spain, Germany, Poland and Czech republic (though some non EU too, not as many) and English is their common language.
My primary age kids go to very tiny local village schools where previously they were the only foreigners, though now there are a couple of others - a Russian boy, a Czech family and a Bulgarian girl.
My eldest is at a large secondary school and just in her class of 30 there are two Polish girls who speak Polish and German equally well, and and two Italian children who speak Italian and German equally well, just as DD is bilingual English/German.
We drive to Italy for our usual summer holidays - it takes a couple of hours, we go through Austria on the way. My kids are not afraid of languages and have a smattering of Italian which they like to use when buying bakery goods in the mornings. We can drive to Croatia too, where DH's uncle and grandma live, without ever having to show a passport. I love that we can move between countries so easily, and that it works both ways.
My "German" PIL are actually Croatian and the descendant of parents from the part of (now) Poland that used to be Germany.
I never felt the need to apply for a German passport, to answer the inevitable question, because I am not German, but I am European. Until Friday a British passport was a European passport (almost, because the UK has always kept its boarders due to never joining Schengen so it has always been far more hassle to get back into the UK than to move around Europe).
I am applying for German citizenship now, because I want to stay in the EU. My kids already have duel nationality obviously. I count myself bloody lucky to have the option, unlike people living in Britian with no grounds on which to apply for citizenship of another European country in order to stay in Europe.
To me choosing just to be British is just foolishly and damagingly limiting - like choosing to isolate any one county of the UK, put up boarders, refuse to allow free movement of people and insisting on visas and trade agreements to move people and trade between one county and another. Setting up a points system will filter people by wealth (and intrinsically linked educational level) instead of nationality effectively ... I suppose then there will be disproportionately more wealthy ex pats in the UK, perhaps that is the aim...
I do wonder where on earth the UK is supposed to be, if not in Europe... The Leave supporters seem to think you cannot be British and European, which is a bit like thinking you can't be Welsh and British, or a Londoner and Englsih.