Old Variac -- chassis ground? is it safe?
I don't see any connections to the case. There should not be any connections to the case.
Assuming there isn't, you should run the ground wire to the case and to the output receptacle. You can drill a hole in the case and attach both to the a screw with a lockwasher and nut.
Measurements made with a multimeter on a voltage range are not reliable since the multimeter is very high impedance and will pick up voltage from capacitive coupling. On a resistance range you can measure between input/output and case and there should be no continuity even on the highest range. Obviously you would do this with power off and unplugged.
You should not connect to either side of the output. A variac is an autotransformer and does not provide isolation. It looks like they are switching both sides of the mains. I think that's acceptable practice today (someone will correct me if not) but you should keep the neutral side (white wire) to the neutral side of the new socket. Modern receptacles  usually have brass for the hot and silver color (eg. nickel plate) for the neutral screws. And green for the ground.

Edit: Well that ohmmeter photo you added is fairly convincing. I have to assume that the plug is polarized with the neutral pin wider- that would be a death trap of a product if not, and even so not all outlets are wired correctly
Okay, looking at some old data (this thing is perhaps 70+ years old)  the diagrams actually show earth connected to one side of the line. This was sometimes done in radios of the all-American-five vacuum tube vintage but usually there was little exposed metal on them.

I would be inclined to find that connection and remove it, and ground the case.