There’s a big difference between American Federal law where the Feds are in charge of enforcing Federal law and the States are in charge of enforcing their own laws; and Pan-EU laws where a law is really an agreement to pass a bunch of laws by the member nations who then enforce them using whatever enforcement mechanisms they have. By passing the agreement as laws through their own legislatures, they in effect become a part of each member nation’s national laws. As far as I know there’s no EU FBI/DEA and the closest agency to it is more like Interpol.
See also the kerfuffle between Apple, Ireland and the EU where the EU went after both Apple and Ireland to get the Irish to enforce some European agreement that may or may not have prohibited whatever Irish tax regime was (is?) in place.