Not that they were necessarily the same, but it seems from what I can tell people had allegiance to and considered themselves to be from a specific state rather than being an American. For example Robert E. Lee's statements and reasoning for choosing Virginia over the country. In this context the ideas of federalism and the autonomy of individual states makes more sense to me than it does today when we're more interconnected than then.
I would say no.
One could perhaps argue that under the Articles of Confederation the United States was a "loose federation", especially as the Continental Congress at that time had no tax-raising power of its own, and any change to the Articles required a unanimous agreement by member states.
However, from the very beginning, the United States under the Constitution adopted in 1787-88 and effective from 1789 was by description a union of states organized federally, with the federal government having direct powers of taxation, regulation of interstate commerce, and a federal executive able to conduct diplomacy on behalf of the entire country, and commander-in-chief of the country's military. Furthermore, the Supreme Court of the United States, especially under its Chief Justice John Marshall (1801-1835), helped to delineate the legal theory around federal supremacy, notably in the McCulloch v. Madison case of 1819.
While in the 1789-1865 period the federal government remained relatively small and limited compared to the 20th century, I think this can be overstated. In 1794 Washington raised an army and marched it into western Pennsylvania to put down the Whiskey Rebellion and ensure collection of federal excise taxes. In 1832-33 Jackson threatened to use force against South Carolina - and obtained Congressional approval - after the state declared recently-approved federal tariffs to be "nullified". Unfortunately, both sides could claim some degree of a win in the climb-down from that crisis.
The idea that the United States was some sort of loose and voluntary association of sovereign states is not historically correct, and perhaps most importantly was not believed by Americans at the time until it began to gain currency among certain white Southerners starting in the 1830s. That has its own history and I would direct you to u/secessionisillegal's great answer on the topic here. Of course state citizenship is technically a thing in US law, and in the 19th century many people identified strongly with their home state, but that's not the same thing as considering the United States even then to be some sort of loose confederation.
As for the European Union - it's something of a sui generis body, in that it doesn't really have any great parallels to other institutions. It has elements of a federation, especially in the sense that there is a body of EU law that has the European Court of Justice to make rulings on. But at the end of the day, it is still largely a international organization - its member states provide the funding (as opposed to the EU taxing anyone directly), and there is no strong EU executive (the President of the European Council has some influence, but is largely a chairperson for a council of all heads of government for EU members, which is far from anything like a cabinet). Even prior to the 17th Amendment in 1913, when US Senators were elected by state legislatures, state governments did not have a literal seat at the table in the federal government in the way that EU member governments do in the running of the European Union. Each EU member also maintains its own military and diplomacy, even where there is a common EU diplomacy, and this is a level of independence that is far beyond anything that the US federal government would have tolerated even in 1789.