Speaking broadly, you kind of touched on the issue. The really ardent secessionists claimed that the US was in fact not a country, but an alliance of sovereign states, much like the EU is today. So there was no such thing as an American citizen or a country called the United States of America. Why this logic fails, though, is when you see all the powers granted to the federal government, and which were denied to the states. For starters, the Constitution through the Supremacy Clause made the Constitution and the laws passed under it supreme over state laws. Today this is the legal concept called Federal Pre-emption. The federal government was also given the exclusive power to regulate interstate commerce, to conduct foreign affairs, to mint money, and to assure that each state had a republican form of government. In short, there are numerous sections of the Constitution that gives the Federal government power to overrule the states, so its hard to argue that the states were sovereign and the Federal government was merely a tool of the states.
One other difference worth noting. The EU provided an express mechanism for states to leave. The Constitution did not. Also, as I always point out, its difficult to imagine that a state had a right to leave when South Carolina decided the best way to accomplish this was by starting a war instead of going to court to have this implied right recognized by the Supreme Court. The mere fact they chose warfare over lawfare tells me they knew their argument held no weight and would be rejected by the courts, and thus no such right existed.