Perhaps it would be a good thing if more organizations were prepared to do more to work together on a local level. As to my reference above about Christianity, there are various non denominational Christian groups, it could be a good thing if there were some of this type of group for Aikido students also where people could train together from time to time irrespective or the organization and style of Aikido offered.
I think this happens, but it's below the radar. I don't mean "hidden" by that, but instead that it's based on personal relationships and not formalized. I'm not sure that making it a "group" would serve the purpose that you want, but here I'm guessing somewhat about what that is. You say "do more work together", but what benefit do you see coming from that? There are potential benefits to ecumenicism, but I have a feeling that they tend to happen when it's a thoughtful exercise rather than a haphazard one.