You should have at least two separate 12v power systems in the coach (3 if the genset had its own). Originally, you would have had two 12v batteries to start the diesel engine, and 2-4 more to provide power to the house (lighting and such) when neither shore power nor genset was in use. The genset starts off either house or chassis battery (usually chassis) if it doesn't have its own battery. Clearly yours has been modified, so hard to guess how things are wired now.
The coach was originally equipped with a 40A converter/charger (75A optional) that provides 12v to the house and charges the house batteries when either shore or genset power is in use. The genset does not directly charge house or chassis batteries - it merely provides 120v power to the converter/charger.
Normally there is an isolator to keep the chassis and house 12v system separate, the idea being that you can run the house system down to "dead" and the engine can still be started via its own battery. Apparently yours no longer is isolated. If you have only two batteries, it sounds like both hose and chassis are combined.
You mentioned a battery switch of some sort. Typically there is a battery disconnect which take the house batteries offline. Sometimes one for the main chassis battery as well. Neither battery will charge if that switch is off - nothing is connected to the battery in that case, so no path for charging.
Everywhere I use a multimeter or a test light it all checks out...
Could you elaborate on what you checked and the results? House 12v system, Chassis 12v system, and/or 120vac system?
the fridge works with the generator running, the shore power will not run the fridge.
Is the fridge running on LP (propane) or 120v electric? In the electric mode, the fridge doesn't know whether the power came from genset or shore, so it cannot matter except back at the transfer switch where the source is selected. HOWEVER, the fridge also needs 12vdc whether in gas or electric mode. If somehow you have no 12v power when using shore instead of genset, then the fridge would not work. Nor would some other appliances and the interior lighting, so that should be pretty obvious.