Unlike the USA—which has a really strong executive branch—there is hardly any executive branch in the EU. Instead the executive powers are held almost entirely by the member states. The EU does have a legislative branch (just like the USA) with around 650,000 people per representative. USA has around 750,000 but that number is further devalued by the USA senate (3,500,000 people per representative) which can hold a lot of power over the congress. I’m not aware that the EU has any equivalent.
The scope of the two systems is also vastly different. The USA federal government represents the foreign policy of each of the state, funds a military which answers to the government, finances much of the infrastructure within the USA, etc. In the EU foreign policy is largely held by the member states, there is no EU military, and funding for infrastructure projects is more likely to come mostly from the member state it self, then from the EU. The USA collects federal taxes, while the EU gets their funding from the member states.
Both have a federated court system. The USA have federal criminal courts which I’m not sure that the EU has, or at least not in the same way the USA does.
To summarize: The US federal government and the EU have in common that they both have a democratically elected legislator (both with relatively low representation) and they both have a federal court system. But this is where the similarities end.