It's interesting making that comparison with the EU, where even though a part of that group, I'd assume most people there identify as their nationality (French, English, German, etc.) versus being European.
That's the main challenge the EU has, getting people to identify as European. As the nation states of the EU existed long before the EU itself, people there naturally identify as their nationality, whereas in the US you identify as American first. The great advantage of the US is it was designed from the ground up for that integrated system of government, instead of having to change existing systems later on. Also there's no escape hatch in US law for states to leave the union. Having, say, California secede and become its own country would be unthinkable (no matter how much the west coast states hated to see Trump win).