The issue I see is bikes "break down" a lot more than modern cars do. Especially with regards to flat tires. Plus, there are relatively few cyclist over which to spread the cost of the service.
Companies like AAA work because they have a lot of members but relatively few field calls. Think about it, out of thousands of cars on the road you really don't see that many broken down. Plus, the towing agency typically tows the break down to their own shop, where they charge for repairs thereby recouping a lot of the cost of the tow.
There isn't enough profit in bike repairs to implement a similar model, so the dues would probably have to be too high for it to be viable.