Wrong with this GFCI outlet

GFCI devices work by comparing the current incoming on the hot wire to the current leaving on the neutral wire. If there's a difference, it means electricity is leaving the circuit somehow (a "ground fault") and the GFCI cuts power*. The GFCI doesn't do anything with the ground conductor—in fact, you don't even need to have one at all. One way of upgrading a 2-prong outlet (without ground) to a 3-prong outlet is by inserting a GFCI device upstream or using a GFCI outlet.

So I'm guessing there is a small amount of current in your scenario, and when you touch the white and ground together, some of that current goes into the ground and doesn't return through the GFCI neutral. Do any of you lights have timers, nightlight, etc.?

(BTW I'm not sure it matters for this question but your description of how the wires up is a little confusing to me. LINE and LOAD are opposite sides of the GFCI device, but you use them interchangeably. If you want a downstream device to be protected by the GFCI it must be connected to the LOAD side. Anything on the LINE side is considered upstream and is unaffected by the GFCI.)

The GFCI is shot, the internal circuitry failed, the voltage measured on neutral may be a leakage from the damaged circuit very little current available but that’s probably why you are reading voltage. Why it did not trip the circuit breaker probably same issue the neutral opened. There are several different types of circuits used by the different companies so it is hard to tell the true cause. I do see failed GFCI’s regularly where garbage disposals are connected and less often but still several times a year where a vacuum cleaner is used on the circuit. These motors are very hard on the electronics in my experience.