Could he have the wrong relay or wired incorrectly? N/O vs. N/C?
I hate to speculate because a vehicle this age, (that has had more than one owner ?), could have been modified over time.
However the feed to the fog lights relay is always live, so if he had gotten hold of some unusual relay for this form factor that is normally closed instead of normally open, that would result in the fog lights always being on until the relay is triggered, even with the vehicle and everything else shut off. I think we can rule that out, unless the way to turn them off is to press IN the dash fog lamp button, the opposite of the normal dash button function, and then that would mean constant current through the relay coil with vehicle off, would show up as additional parasitic current draw above the typical ~25mA once the vehicle is timed out from the battery saver circuit and asleep in its (otherwise) lowest current drain state.
Easiest would be just look at the part # for the relay but I doubt that it's normally closed, especially since the vehicle does not seem to have the fog light isolation relay used, so "probably" is the same circuit I saw in the wiring diagram.
If it is the same circuit, the only thing that comes to mind that makes sense is that someone has rewired the high and low beam circuit such that the relay coil is grounding through the high beam bulb element instead of low beam. Unplugging the headlamp connector (right headlamp) and checking resistance between its high beam and fog lamp relay socket pin 1 would verify this, since it should instead have low resistance between only the low beam contact and relay socket pin 1. Wiring diagram I'm looking at is attached.