Look at the list of undertakings in #3471.Why do you bounce the question back? Your claim it's a capitalist institution, you should back it up.
The concept of state exists in capitalist and in socialist societies as you say. It existed in feudal societies and in the slave empires of ancient times. Even Genghis Khan established a state. Thus, it is not true to say that it is not a socio-economic institution; because each and every state has one socio-economic character or another, it is an institution common to most or all economic systems, and each state necessarily reflects one or another such system, be it, say, feudal or capitalist. Therefore although the word state doesn't denote any specific system, it makes sense to refer to capitalist, or feudal states.But I'll try anyway. The EU is a political institution very much like a state, a "super state" if you will. The concept of state is not necessarily a capitalist one; socialists nor communists endeavour to abolish the state (anarchists do, however). So, in my view, "state", and consequently the EU, is neither a capitalist nor a socialist institution.
Let me put an analogous point. The word "hat" does not denote any specific colour of thing. One can have a blue hat or a green hat. But every hat possesses one colour or another. So although the word denotes no colour, no hat is without one colour or another as one of its specific attributes. Thus, your "(the state) is neither a capitalist nor a socialist institution" must be amended to read "may be either a capitalist or a socialist institution". Or indeed some other, as observed. The EU may or may not be a state, but it is unquestionably a capitalist institution.