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.

You mean like Strasbourg?

https://en.wikipedia.org/wiki/Seat_of_the_European_Parliamen...

Spoiler, the parliamanet moves once a month between Brussel and there. That's how centralized the EU is, we cannot even decide on one fixed place to meet and decide.