> The state and federal governments are both “sovereign,” and the states were there first, exactly like the EU.

The majority of the US states were not there first – the majority of the US states were effectively created by Congress. Federal territories–which as such were fully under control of the federal government–were elevated into states by Congressional decision; frequently, Congress also decided how many states some federal territory would be partitioned into. The only states which could be said to have had some independent sovereignty pre-existing federal control were the original 13, Vermont and Texas. One might also count Kentucky and Maine, being consensual splits from the original 13, in the list; possibly also West Virginia, although the degree of consent involved in its separation is more debatable. So we are talking about 15/50, maybe 18/50 if we are generous. (Hawaii doesn't really count because the State of Hawaii is not continuous with independent Hawaii, since for over 60 years in-between Hawaii was a federal territory–but even counting Hawaii, that's only 19, leaving 62% of US states created by Congress out of non-sovereign territories.)

It is also different in that US states can have their sovereignty further limited, against their consent, by constitutional amendment (which they may have voted down, but the other 3/4th of states supported), or by Supreme Court decision (who decide what "state sovereignty" means in practice, and can choose to interpret the concept broadly or narrowly, and may alter its boundaries over time as their case law evolves due to changes in judicial whim and the ideological composition of the Court.) By contrast, under EU law, any further major transfers of sovereignty from the member states to the EU require unanimous agreement.

And the biggest difference is that the EU is a club you are allowed to leave (as Brexit demonstrates), the US is a club from which exit is near-impossible (see the US Civil War). That fact gives EU member states vastly more sovereignty than US states have. Indeed, it is disputed whether secession of a state from the US is possible even if Congress agreed to it – many say it would require a constitutional amendment. The EU treaties now contain an explicit right to leave, and it was generally assumed that one existed implicitly even before it was explicitly stated. If the EU tried to reduce the sovereignty of a member state against its will, bypassing unanimity (such as via qualified majority voting, or via an ECJ decision), the member state can always respond by leaving – and it is likely a mere serious threat of leaving would result in them being allowed to stay but with an exemption from the decision in question. What US state could ever hope for that kind of sovereignty?

I don't know why so many Americans want to compare the US states to the EU member states–it is not a very sensible comparison. Better comparisons for the US states would be the Canadian provinces, the Australian states, the German Länder, etc–they are roughly the same kind of thing as US states are (units of a federation lacking their own sovereignty under international law), whereas EU member states are a very different kind of thing (fully sovereign states under international law)