well, I'm Dutch, too, so I can't really give an answer to the state stuff, but I can guess that most Americans consider themselves plain American, and not necessarily distinguish one state from the other in the big picture. The whole "American and proud of it" stuff is shining out of every hole in the American culture, and that is something not many countries have. I mean, how many times in the Netherlands do you see the devotion to the country that is anywhere near the zealotry I've seen in the states?
I suppose cultures can clash in between states, and rivalry might be around, but in the end it's all pretty tight, and the American people are proud as hell of their heritage.
To return to your original question, I'm Dutch, and will never say that I'm European when I am asked where I'm from. Comparing Europe to the US is silly, as it isn't one country with one government. And yes, I do have a sense of nationalism in a province-city kind of way. I will never lower myself to the point of saying that Rotterdam is the same, or even better than Amsterdam, I'm not stupid (no offense if you live in Rotterdam, though).
Amsterdam-Dutch and damn proud of it.