There used to be a lot of customs unions, currency unions and so on in Europe. Many of those were absorbed by the EU, but surely there are others that still exist. Another way of looking at this is, should the EU disintegrate what would the next largest trade or economic organization in Europe be?
To be clear, in order to answer this question from my perspective the organization:
Can include EU members, but must be governed independently of the EU.
Include primarily or exclusively European states, but if it has a few debatable or explicitly non-European states, that is fine (e.g. Iceland, Cyprus, Israel) as long as they are states which have strong European ties.
Be related primarily to trade or economics, such as a customs union or a shared currency.
No EU country may enter economic treaties with other countries on its own - everything has to be done through the EU. That's the core principle of the Single Market. Within the EU there is (nearly) absolute freedom of movement for goods anyway, while all goods entering the Single Market from outside the EU have to adhere to common rules.
So the answer is simple - no such organization can exist by definition. Either it includes the entire EU or it cannot function in any meaningful way.