I recently happened to stumble across a stream of a game called "The End is Nigh", and I noticed that the background music in that game consists of remixes of classical music (for example, "Marche Slave" or "Danse Macabre"), and I am curious what other games have taken a similar approach.

So, what other games are out there that base their soundtracks on public domain classical music?

Lemmings used mostly public domain music for its soundtrack, as I recall.

Not sure if it's public domain, but the Japanese eroge Crescendo's soundtrack is what comes to mind for me...

From Wikipedia http://www.youtube.com/playlist?list=PL4046E3FDA970ECDB

For PC : Bioshock 1, Bioshock Infinite, Earthworm Jim, Elite, Frontier: Elite 2, Lemmings (inc Oh No More Lemmings), Lemmings 2, Loom, Psychonauts, etc. Note : Many of these have only 1-2 pieces in, not the whole or majority soundtrack.

Bigger list here includes console games:-
https://allthetropes.org/wiki/Public_Domain_Soundtrack#Video_Games

Not a game but I swear every single Warners Brother cartoon has public domain classic music behind it.

Like this one:

https://www.youtube.com/watch?v=M-52C6KSeM0

Mount and Blade Napoleonic Wars

https://www.youtube.com/playlist?list=PL6FC7C8E93E35EB64

Wishbone beat me for Lemmings which had some tracks based on classical music (because it was free).

There was a sequence in Full Throttle where the Ride of the Valkyries was played. There must be tons of games out there which include such sequences with specific classical music.
Talking about the Ride of the Valkyries, Aces of the Deep (1995 subsim) had it along with a dozen others as available background musics.

Alone in the Dark also had 3 musics as ingame items you could play with a gramophone (the Beautiful Blue Danube, another one from Chopin, and the Macabre Dance from St Saens).

I also recall Maupiti Island, where you could play a dozen tracks on an automated piano and watch the actual keys being pressed while playing.

There was also Gabriel Knight 2, which had a soundtrack heavily inspired by Richard Wagner, while not technically being classical music since it was composed by Robert Holmes ; it sounded really like it at times.