Ok so...
Yes I have a built in blow dryer in the bathroom, it helps keep my luscious locks looking good!
I only have one known light in the bathroom, built into the ceiling. There is no light over the shower, instead I have a large domed sky light.
I looked up my water heater heater model and it is propane only. Even so I checked in the water heater compartment and there is no secondary switch there.
I turned the fan on then disconnected the switch but the fan stayed on so this tells me the switch is not connected to the fan in any way.
I then checked for voltage and found that the circuit appears to be dead. I was getting no reading at all from the wires with a voltmeter. This switch is part of a two-gang switch and I checked the other switch and it was reading 12V. Because it is the same style switch, same connectors and same gauge wire I would assume that the switch in question is also 12V but I can't be 100% sure.
I checked all my fuses and circuit breakers. All the fuses were good and no breakers were switched.
I think it is likely at this point that this switch controlled the fan at some point but when the circuit went down it was bypassed and the fan was wired directly.
However, the idea that there is a built in light in the medicine cabinet somewhere is interesting. This trailer does have "mood lights" in various places and I have found small lights on top of other cabinets. It is possible that there is a small light either on top of the medicine cabinet or inside it somewhere where it can't be seen easily, which has a blown bulb.
The trailer is back in storage for now but I will pull it out next weekend for a trip. I will take a look then.
Thanks everyone!