They print them off at the restaurants. Most Disney restaurants can add "Happy Birthday" or "Happy Anniversary" to their printed menus on request, and possibly names but I'm not sure.
. . . correct
. . . it is not elaborate, but still a nice touch
. . . ask at the podium when you arrive
. . . you can list an event, but not request a menu via ADR ressie
. . . ANY sit-down eatery can do a personalized menu
. . . however, if the CM's don't have time, they won't do it
. . . it is totally up to the CM's at the eatery
If you're asking about menus arranged for beforehand with photographs - the kind Coral Reef used to do - I don't believe any restaurant does that anymore.
. . . correct, again
. . . adding pictures got just too cumbersome
. . . too many people read about it and wanted it
. . . when there wasn't enough time, then they complained LOUDLY
. . . it was no longer the time, effort, or risk of continuing the practice