The EU also theoretically has no real military (yet) so right now NATO is the boogeyman they use even though the EU has undoubtedly also been creeping toward them in the same exact way NATO has. Also a large amount of forces that would make up a unified EU are also the countries that are NATO members.

EU and NATO cross membership also backs them into a corner. They cannot aggressively fight the EU off without setting off a war with non EU members of NATO like the USA and the UK who would be called in any defensive war against a NATO member state, but if one day the European continent unites and has an aggressive military there is no such mechanism where the United States and UK would be called to their defense.