I was born in Europe, I spent the first 33 years of my life in Europe (and, given I am not quite 51 as of this writing, that really is most of my life), yet, I never really thought of myself as a European until I moved to the US.
I mean, I am from Europe, so that does make me a European, but it just never was the way of identifying myself. Europe is not a country, Europe is a Continent. There are many countries in Europe, many nationalities, many languages, many different cultures.
Take, for example, Bratislava, my home. It is a city in Central Europe. It is the capital of Slovakia, located on the Danube. People there speak Slovak, a Slavic language.
If you go up the river, you will soon come to Vienna, the capital of Austria. People there speak German (or, rather a unique dialect of German), which, of course is not a Slavic language. Both Slovak and German are Indo-European languages, but speaking either one does you no good once you cross the border to the other country. Yet, the two capitals are only about 50 kilometers (about 35 miles) apart.
If, on the other hand, you sail down the Danube from Bratislava, you will soon arrive in Budapest, the capital of Hungary. People there speak Hungarian, a language that is related neither to Slovak nor to German. For that matter, it is not an Indo-European language.
Having grown up in the very center of Europe, I then never thought of myself as a European. I have always thought of myself as a Slovak. I would never deny I was a European. Had you asked me back then (or even now), "Are you a European?" I would have said "Yes," without hesitation. But if you asked me "Where are you from?" I would say "Slovakia."
The first time I ever heard anyone grouping me up with unrelated cultures was when I lived in Rome and when my American friend's parents came to visit. They had been in Italy once before and were, therefore, constantly commenting on how "Europeans" act and behave.
For the first time in my life did it occur to me that, wait a minute, I am a European, and I do not act anything like what they were describing. I made that point very clear to them, and they stopped talking like that.
The truth is the typical Italian looks at the world and behaves differently than a typical Slovak, yet differently than a typical Englishman, or a typical Swede, etc. I am not saying the typical Italian is better or worse than people from other European countries, only different.
That said, one might say there is no such thing as a European, at least not in the sense people in the US often use the term. Of course, the term European is still correct when simply referring to someone born and raised anywhere in Europe, or perhaps someone born somewhere else but moving to Europe permanently. But it really should not be used to make any general observations about "how Europeans behave" or "what Europeans do."