Did you check the earth to the socket?
If there is voltage between, say pin 5, the right rear light, and 3, earth of the socket when the lights are on, and putting 12 V from a battery on pin 5 of the plug with the -ve of the battery on pin 3 of the plug, then it's got to work.
Things that could be causing a problem include:-
Poor power connection to the bypass relay, so that it can provide 12 V when the only load is a multimeter, but it can't provide enough current to power a light. It's possible that the bypass relay will provide around 12 V when there is no 12 V power feed to it, but the corresponding light is on. However, as soon as a load is applied, the voltage will drop to nothing.
The wrong earth connected on the 13 pin socket. Pin 3 is the earth for the lights. The other earth pins are for the power feeds that aren't used on a bike carrier, but someone could have been using the other earth pins and pin 3 isn't connected.
Bad connection to earth on the socket.
Whether the car has CANbus has little to do with trailer lights. That is an often repeated myth. The biggest thing that associates them is a general lack of understanding of the blown bulb detection circuits in cars, and of CANbus.
On just about all cars, the light clusters have individual wires for each function. Some cars will detect the current taken by the light clusters, so will be upset by adding trailer lights, or will have bulb testing systems that cause LED trailer lights to flash. That is why bypass relays are needed, which detect the voltage being sent to the car's cluster, and provide the same voltage to the trailer lights with the current for the trailer lights coming from a separate feed. Some will delay the trailer lights coming on by a short time so that short pulses used to detect the car lights won't flash the trailer lights.
Sometimes the module that detects the current taken by the lights will send a signal to the the instrument cluster using CANbus, but it could be sent using wires, LIN bus or Flexray. An association between CANbus and needing bypass relays is that sensing lighting circuit currents and CAN bus both became common on cars in the 1990s. Some cars have trailer lighting modules that are connected to the rest of the car with a CANbus, but however the signal get to the trailer socket, the final signal is 12 V = on, O V = off for each light.