Tram
It could be, the split relay coil circuit is confined totally to the car, it relies on sufficient pull in volts to energise the relay, the relay makes a contact close, which in turn allows the fridge supply through to the tow bar sockets.
When the car engine is switched off, the battery volts quickly decay, at some point the drop out voltage is reached, and the split relay coil circuit is de-energised, the supply to the fridge cut off.
As sir roger says, do not overlook the "earth" connection, this is usually a short wire to the cars body work.