Three types of power supply.
1) Simple transformer and rectifier switches off on 100 times a minute, should not be left connected for extended time or used without a battery.
2) Switch mode power supply again can have no smoothing between off and on, and often monitors battery voltage jumping between stages as it charges, could do all sorts without a battery being connected. Often called a stage charger.
3) Again likely switch mode but with smoothing and one fixed voltage output, designed to replace or maintain a battery, this type does not need a battery.
Types 2 and 3 may be combined in one unit with a switch float/stage this should be in float mode with no battery.
In the main switch mode supplies have auto resetting over load,
As to fridge from car, in the main the caravans battery is completely independent from battery charging, this is laid down by BS7671:2008 not got new version but not heard of any changes.
However this is a British standard Germany may not have the same regulations, well German 220 volt sockets are not permitted in this country so clearly the regulations don't match.
The main point is there needs to be a method to stop the caravan battery trying to run the cars starter motor, for years this was done with a voltage dependent relay, but when cars started switching off alternators when under load and stopping the engine at junctions.
Until the automatic anti snake device it really did not matter if the car supply switched off, however today loss of power from car could mean loss of anti snake device.
So standard was two supplies car to caravan one all the time and one only when ignition is on, the relays in the caravan turns that into three supplies as the ignition supply works a relay so permanent supply is switched on when ignition supply is live.
It seems some German cars don't connect both supplies, however since we don't have a German car I am not sure which don't as standard have both supplies.
Clearly if the German spec does not have twin supplies, then testing it and it not having second supply would not be seen as a fault.