You have passports and checkpoints and all that stuff in the European Union.
When I get into a car and drive from here through the Netherlands and Belgium to France, I would not have to show my passport even once. I wouldn't even drive through any checkpoint at all. Our local border to the Nether lands for example is for the most part very hard to notice, it's just that suddenly the roadsigns are different
But of course, all those countries are still sovereign, independent countries. There's no real EU government. And as far as i know there ain't no Texan ambassadors to to Ohio or North Caroline either. USA, as a sovereign nation is a UN member, the EU is not - because it ain't one.