The EU is not a republic. A Republic is a state in which political power isn't hereditary. The EU's power however comes from a coalition of countries, most of which are monarchies.
Contrast with the US, which is a Republic:
- The US has an elected government which is in charge of the executive. The EU is made up of a mixture of institutions, some elected, but the elected components do not have control of the executive, that's left to the member states.
- The member states of the US are Constitutional Republics. Most of the European member states are Constitutional Monarchies.
Even if you'd argue "Ah, but the UK is a monarchy that's secretly a democracy", those concepts aren't contradictory, the UK has a monarch who has deigned to allow the people to determine the membership of the legislature, but that doesn't mean that it's not a monarchy. It's just democracy turns out to be compatible with all kinds of constitutional systems, just as Pinochet's Chile, a Republic, or Hitler's Germany (after 1934), a Republic, proves Republics aren't always democracies. Democracy requires those with constitutional powers allow the people to vote, who knew?
Anyway, the point is it's not a Republic. It's a Union of different groups that means it doesn't fit the strict definition of either Republic or Monarchy. It's just its own thing.