Is there a setting I need to change to make this aircraft ID quiz show up in the game? I remember playing this on my PC when I was younger and I don't remember this quiz showing up then either, but the manual mentions it.

Aircraft recognition quiz was there mainly as somewhat poorly disguised copy protection, which, by my own recollection failed very miserably with players of my age back then
(if you were a fan of combat flight sim in the early '90ties, chances were, that you knew all NATO and Warsaw pact combat aircraft very likely anyway and the lack of manual made you just chuckle and move along...).