> 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

These types of EU laws are called EU directives, which as you described first need to be "transcribed" into national law by the respective parliaments. As a result they don't go into into force in all member states at the same time, and there is some more leeway concerning their implementation.

However, there are also EU regulations, which are immediately binding in all member states in their original form as soon as they are passed by the EU, much like US federal laws.

It is true though that even for EU regulations, the EU generally does not have their own enforcement capability and so there can still be differences from state to state on how strictly regulations are interpreted.

The GDPR (General Data Protection Regulation) is an example of the latter kind of law, btw.