I had some concern about this prior to our last trip, because everything I read insisted that policy was to always ID everyone, regardless of age, and that a passport (or copy) was absolutely required.
We had a dining plan. My mother, husband, and I (age range 40-62) each had a drink with dinner every night of a 7 day trip. None of us were ever asked to show ID of any kind.
Like all things Disney, it's probably a YMMV thing. My guess is that for those of us who can no longer reasonably be mistaken for a teenager, being asked to show ID is possible, but rare. And the insistence on a passport would be rarer still.