I have never heard of this happening at WDW.
Disneyland - yes you need it to enter the 1st day, but I've been on these boards for many many years, been the spouse of a military member for over 20 yrs. and have used different forms of military tix over the yrs. We have NEVER nor have I ever heard of anyone ever having an experience like you're reporting at WDW. I would be beyond POd if something like this were to happen and I just cannot imagine Disney ever even trying this. None of us carry our ids to the parks...I would find it absolutely absurd if a CM asked for my military id in the middle of our trip at the park and cannot see it happening. OP, you will be fine.