MIDI songs? I checked, I couldn't find any from the link you posted. Most were different module formats, like XM, Protracker, S3M, Impulse Tracker. Those have nothing to do with midi other than they also produce music.
I'm still since then always assuming the above when someone says "MIDI music"; they really mean "really basic/simple music" or just straight up "chiptune" sometimes.
It has nothing to do with MIDI really, just a misnomer.
The same behavior could be (also was) teased out of a MOD player if you choose samples with a handful of sample points, like 12. You could also draw up a sawtooth in paint and use that as a sample. These are down-to-earth honest true Scotsman chiptunes.
[0]: Soundtracker, the tracker that started this all: https://en.wikipedia.org/wiki/Ultimate_Soundtracker