Not so. US states cannot have a standing army without consent of Congress. The Congress has given such consent in the form of the Militia Act a century ago, which enabled the states to create the National Guard and the State Defense Forces / State Guard.
And while NG can be federalized (i.e. put under the federal chain of command, with the president as commander as chief), SDF are not - their chain of command terminates with the governor, and they cannot be used outside of state boundaries.
Most people in US know about the National Guard, but not about SDF. It's largely because the latter is rather small and obscure, and mostly used as a sort of civil defense agency with a military chain of command (many states don't even arm their units). But it's an interesting system: https://en.wikipedia.org/wiki/State_defense_force
EU states are nation states, US states are not. The EU is not a nation, the US are.
EU states have much more sovereignty than US states written inside laws, but more importantly, through support of their people. This is why there is a Euro crisis and why Brexit happened: The unification of Europe is a beautiful idea and I personally hope it will recover, but while subsets of it were considered as one nation at some point (Austria, Germany etc.) the EU as a whole simply never was and never will be.
The EU, despite its name, is essentially a humongous cooperation deal between countries. It has many of the trappings of a nation, but that's an illusion. EU countries are sovereign (consent is required in any alteration of the membership rules and they can leave at will) but US states are not. That's the theory. In practice, the differences are even bigger: There's no EU patent system, for example.
I think the EU is becoming (and in some senses arguably already is) a quasi-state. Hence Brexit. That's kick back.
And there is now an EU-wide patent system: https://www.unified-patent-court.org/
Even that is a little shaky, as historically there were organized state militias, which eventually evolved into the National Guard. Even now, the National Guard formations for each state are under the direct control of that state, rather than directly subordinated to the federal government.
It's very much like the EU: Each state is independent, and there is an overall governing body - the federal government, which for the most part can't tell states what to do, it can only give or withhold money to induce them.
There are specific rules about what the Federal government can and can't tell a state.
One difference that does exist is military, where EU states can have a military (as far I know), but US states can not.