Brake lights ground where?

There should be only 2 wires at a rear lamp socket. Brown is park lamp circuit. Green or yellow for left and right are both brake and turn. Yes the brake lamp current could back feed through the park lamp circuit.
It would attempt to power both park lamps and the tag lamp. Maybe not light'em up brightly but at least show power on a meter.
You probably just had a weak ground connection at that other ( dim ) socket.
I don't read a actual fault at this point but for what its worth... Whenever I look at a fault in rear lighting, the 1st thing I look for is where trailer lighting may have been tapped in. I've seen some pitiful hack jobs over the years.