TLDR; a Spaniard can say I'm not Spanish, and a Dutch can say I'm not Dutch, but neither of them can say I'm not European, since their experience as Europeans is as valuable as mine.
It's part of an identity. I could mention history, culture, religion, linguistics, art, influence, heritage, race (?)... I speak four European languages and have noticed that, Europe-wise, there isn't all that much that divides us.
I should perhaps mention I am not born-European. I am, in fact, (white) Latino.
I currently live in Spain. I have two European nationalities and one of them is Spanish. According to European laws and Spanish government, I am Spanish. But in reality, I'm not considered as such. Spaniards are Spaniards, Dutch are Dutch, German are German... These peoples have very deep, basically tribal histories with each other and their land. Their identities are not based on their system of government. They know who they are, and they pragmatically try to govern themselves in whatever way seems best. All I am to them is an expat/foreign/immigrant. It doesn't matter if the immigrant has lived all his life here. The most he'll ever be is practically German. But never fully German, or Spanish or English or Dutch. Just because you have our nationality doesn't mean you're one of us. That's why when we see those ISIS terrorists and the title refers to them as French or British or Dutch, one of the most upvoted comments will be He's not and will ever be French in a vulgar speech.
Same applies to me too. I've lived among the autochthonous for many years and speak exactly, with same accent and body language as any of them, yet my skin/religion/name/place of birth/facial structure will always make the difference between being one of them and a foreigner.
I understand that and I am perfectly okay with the fact I won't ever be considered one of their own. I embrace it yet I am not ashamed of my origins and background. No law will ever be able to break that.
However, I have noticed that the European Project is (intentionally or unintentionally) slowly setting up a European mindset. Which gives me the freedom to integrate myself and compare myself among other Europeans and not just among Spaniards or just among Dutch.
I'd say the EU is slowly turning to a USA construction. In which the shared experience of citizenship in Europe is based on political and cultural conventions to which every European chooses to dedicate themselves.