I think you're interpreting that backwards. "Ludus" originally referred to educational training sessions to build skills for later performing activities of direct practical benefit, mainly combat and political oratory. This is how "ludus" means both "play" and "school." It's originally "play" in the sense of "play-fighting," not in the sense of "having fun without a particular goal." Unlike Greek academies, Roman schools were entirely focused on building practical skills.
The idea of "play" as "having fun without any particular goal orientation" is a thoroughly modern idea. The idea that school should be a form of "play" in the modern sense is an even more recent one. It has no basis in ancient Rome.
So our Latin teacher lied to us! I should have known...
Thank you very much for the explation!
EDIT: In Latin class, our textbook was called "Ludus Latinus", and our teacher made a big point of explaining how that meant learning was supposed to be a kind of game. It made me groan at the time, but in later years, I always thought it made perfect sense.
I'd translate "Ludus Latinus" as something like "Latin Practice," in the same sense you'd say "soccer practice" or "piano practice" or something like that.
It is a game, in the sense that it's not the real thing, it's practice for later doing the real thing -- it's a structured, goal-directed game with a particular practical outcome in mind (building Latin skills), not a "let's go have fun in the park without any particular motive other than having fun" kind of a thing.
The idea of "play" as "having fun without any particular goal orientation" is a thoroughly modern idea. The idea that school should be a form of "play" in the modern sense is an even more recent one. It has no basis in ancient Rome.