As the saying goes, “A rose by any other name would smell as sweet”
EU law and its constitution have primacy.
EU law and its constitution have primacy.
But mostly, EU law just sets a baseline, and almost all execution of it is devolved to the member states.
Edit: the EU does not have a constitution but a constitution shaped update of the Treaties. Now, lots of politicians are happy to blame the EU for unpopular stuff, but the council is the national politicians and they basically run the EU.
The EU is not a state. Personally, I'd prefer a much more federal EU, but that won't happen anytime soon.