These routes aren't technically "franchised" or "subsidized" by NYCDOT, but the only thing EDC knew about buses was the fact that they had wheels and carried people. In New York City, any non-MTA entity receiving state money must have it passed through NYCDOT because they are the state agent for that program downstate.
NYCDOT provides an immense amount of support for these routes, including using Green's schedule format to publish schedules for the X23 and X24. The fareboxes are owned by the City of New York as well, and the MTA refers any non-MTA carrier that wants fareboxes to NYCDOT. They also monitor those services periodically and while these two routes aren't official franchises, they are treated as franchises, which triggered the City Council uproar.
By the way, the plan is for the MTA to take over the routes at the end of the three year contract, but this may not happen because they have nowhere to park these buses. Then it would have to be rebid. This is why they wear the X23 and X24 names instead of their original names, which were AE1 and AE2.