Your digital voltmeter was measuring a minute amount of current, enough to make the meter run but not enough to run a light bulb.
With the GFCI bad there could be internal carbon traces or even a high valued resistor that does some sensing that is now feeding back power.
If you had left the light in the ON position there would have been a load on the line and your measurement would have been zero volts.
To demonstrate my theory, put the meter in the DC volts position and go to one of your 12 volt batteries. You could do this with a flashlight battery too.
Place one of the negative leads of the probe on one post of the battery. Now with one hand pinch the other lead. With your free hand, touch the other post of the battery. By pinching harder or wetting your fingers you will get the full battery voltage showing. I promise you there will be no shock or any sensation at all.
