Your Fridge's interior light should operate if you do have 12-VDC at the socket, and if the door light switch is operating properly then all should be well.
One thing that happens to some people is the COACH battery may not be fully charged, and thus cannot provide an actual 12-VDC at the bulb when there is a load on the battery. But, if your Rv's interior lights operate OK then this is not likely to be your problem.
This leads to two probable situations; 1- the Bulb is bad? or 2- Your ground wire to your socket is open and you would need to trace this back to the rear of the Fridge.
PS. Make sure you measure your 12-VDC relative to a good ground connection.