I don't think it should go to an institution whether the institution is the EU or not. Seems a bit daft to me and not what the prize was intended for.
However doesn't everyone know that a big reason the Common Market, (which evolved into the EEC and then the EU) was set up was the political desire to avoid another European war? No one can deny it has achieved that so I assume that is why it won the prize.
As I said though giving it to an institution of any kind doesn't seem right to me.