For most people it's obviously not an either-or thing, being European is a part of their identity like their nationality, the region or state they live in, their city, etc. Which part of their identify comes to the fore most depends on the context - if it's about world politics, they'll probably think of themselves European, but if it's an international football game, they'll think of their country first, and if it's a domestic league football game, they'll think of their city first.

The ones who feel the European part most strongly are mostly the people in smaller open nations without a strong nationalism of their own, such as Belgium or the Netherlands, or those who identify much stronger with their region than with their country (many Catalans, Flemings, Scots and so on figure that they should just become independent states within the European Union, by getting rid of the in-between level of their countries Spain, Belgium and the UK respectively). Or people from countries that have only relatively recently joined the EU, and can still remember living in much worse conditions before that, like those in Eastern Europe (but also Spain, Portugal and Greece).

And there's an age factor - young people who are more likely to speak foreign languages and to have spent time abroad (especially if they are highly educated), like in the huge "Erasmus" foreign exchange program that one hundred thousand or so of university students participate in each year, are obviously more likely to identify with their peers in other European countries.