We don't know what his continuity reading actually is. If he has DMM with a continuity beeper setting, many those only beep when continuity is less than say 50 ohms. If he has a higher resistance than that and is still calling it continuity, which isn't exactly an incorrect statement, he could easily have readings like he has on the plug in polarity tester.
I think it would be best to check voltages with a low impedance meter, if something has a higher resistance to it then it will likely show up as something not expected in those readings.