Nowadays, that thinking seems wrong. Not sure what yall think over there, but I've always thought of Michael Jackson an American artist, rather than a person from Indiana. I just don't see that Americans see the states as a federation of countries, but I guess people there might.
Something like the UN might have similar votes per country, on the rationale that countries are equal regardless of relative size.
The US when it formed was probably more aligned with what the EU is today. That original setup was very impractical so they reformed into the current US with the Constitution in 1789.
There was an episode during the ratification debates where Rhode Island was almost left out of the United States. 12 other States had already ratified the new US Constitution but Rhode Island held out for more concessions until eventually it was decided that Rhode Island could ratify or leave the Union. Considering they were surrounded by much larger States who had prior grievances with Rhode Island, Rhode Island made the right call. Otherwise they’d likely be part of New York or Massachusetts right now.