I think the EU project is partially trying to thread the needle of maintaining ethnostates without the nationalism. That is, maintain the culture/language/food without the periodic bouts of war/etc.
Neat idea, but is quite a challenge.
The US in general is different from even an imaginary future federal EU in that, as some say, "America is an idea". I know people who can trace their "American Heritage" back to pilgrims/boats, and others who naturalized in the last decade. No one aside from true nutcases really gives much thought to the difference. There is no legal distinction aside from naturalized citizens being unable to be President, is about it.
Most people I know have lived in multiple states, sometimes across completely different regions. People don't derive any strong identity to the state they were born or spent the most time in. Most people I know would just identify by the current state they live in.
There are of course regional differences, but you can move from one region to another and mostly drive the same car, eat the same food, wear the same clothes, buy the same products, shop the same stores, etc. This is a feature or bug depending on your perspective. I think we are sort of like a cultural Borg.
The plus side is the US quickly adapts and/or adopts aspects of the culture & cuisine of each wave of immigrants over time. You can get pho in random strip malls nowhere near a big city, and we'll adopt any holiday if it means more drinking, like say Cinco de Mayo.