> but when we travel between states we all use the same money

I think you're significantly exaggerating how close-knit the EU is. Its member states are generally more closely aligned on economic matters than in other areas, but not all of them even use the same currency. Currently, the Czech Republic, Hungary, Poland, Romania and Sweden all have their own free-floating currencies, while Bulgaria, Croatia and Denmark have currencies that are loosely pegged to the euro. In areas such as criminal law, social policy and national elections, the EU has basically no say whatsoever and the member states are free to do what they want. The member states are also free to leave the EU, and their national governments have a much bigger say over the EU's decision-making than US state governments have over the US federal government.