The culture in the US varies some from region to region (midwest vs northwest vs south etc), but it's absolutely false to say that the differences are "vast", especially on the state-to-state level. Standards of living vary much more between urban/rural even within states than they do across states generally.
Leaving aside the fact that EU states speak completely different languages, the cultural differences there are much, much more pronounced. Everyone in the US speaks the same language, watches the same TV shows, follows the same sports, shops at the same chain stores, etc, etc.
Your comparison of the political structures is also quite off, though not very relevant to the question of cultural homogeneity.
There was an interesting discussion of this and related topics here just the other day: 'Duck Dynasty' vs. 'Modern Family': 50 Maps of the U.S. Cultural Divide (nytimes.com) https://news.ycombinator.com/item?id=13264083
I moved in the last year from a mid-sized city in the northeast to a sightly smaller mid-sized city in the south, and everyone keeps asking me excitedly about the culture shock. People eat more okra here and it's hard for me to understand the speech of the few people I've met from more rural areas of the state. That's it. The culture here is "mid-sized US city," just like where I came from. It's almost disappointing.
This sidesteps the question of cultural differences state to state / country to country, or even between urban/rural regions.
Sources: https://en.wikipedia.org/wiki/Comparison_between_U.S._states... https://en.wikipedia.org/wiki/List_of_countries_by_GDP_(nomi...
Bulgaria: 6,843 USD per capita; Mississippi: 24,062 USD per capita
Bulgaria is a member of the EU. If you look outside the EU (but still in Europe) you find countries that are poorer than Iraq and Vietnam.
https://en.wikipedia.org/wiki/List_of_countries_by_GDP_(nomi...
I'd argue they are mostly the same. IMO, The US is pretty similar in structure to the EU where the US as a whole is comprised of many states, each with their own laws. Yes there are federal laws but for the most part, each state can do whatever they want.
Each state / region has vastly different cultures and standards of living.