No, it is not quite the same as Moscow and Washington are capitals of centralised states who give orders to the whole nation.

The EU on the other hand does not have a common constitution, army etc. so is not a real state (yet). It is made up of soveraign nations who come together debate and decide there, but then it is still up to the members to implement that.

So the transition to the EU as one state is happening, but might never complete.

The European Commission is in fact empowered to boss member states around, it's one of the things that give EU law teeth rather than it being like "international law" (unenforceable anarchy). It also acts much like a government (in the sense of executive, not in the sense of state) when it comes to EU lawmaking, and has various government-like powers in fields like competition law for example. And the European Commission is based in Brussels. Saying "Brussels" to refer to Commission activity is as natural as saying "London", "Downing Street", "the Cabinet Office", "Whitehall" etc to refer to British government functions. And that's without getting into all the other EU institutions that are based there!

It is true that the EU institutions are ultimately subordinate to the member states in a way that, say, the US federal institutions are not, but the EU is still very much is its own thing. It even has legal personality these days: you can sue the EU and the EU can sue you.

It doesn't imply that the EU is one state. It's just the place where the decisions are made. If Brussels didn't like anyone knowing that, I'm sure other cities in the EU would happily take the gobs of free money showered on wherever the EU is headquartered.

I’m not sure you realise that this is a far more generic rhetorical phenomenon that encompasses all kinds of situations. Like referring to the FBI as Quantico.