If we want to get technical, MIDI stands for "Musical Instrument Digital Interface". It's a protocol for transferring data. Often musical data. It's kind of like sheet music turned into a stream of numbers. MIDI itself makes no sound. To get sound from it, you use MIDI data to trigger either a musical instrument that accepts MIDI input, or a wave table, which is a collection of sounds typically designed to imitate real instruments.
It was used in games because of limited storage space and computing power. Raw audio files take up a lot of memory. Old consoles and computers didn't have a whole lot of memory. Playing those same files takes a lot of computing power. If you want decent sounding audio, you need something spitting out samples at 44.1 kHz. That's 735 times faster than 60 Hz: your typical frame rate.
MIDI data takes up much less space because it's not raw audio. It doesn't handle every sample of 44.1 kHz audio. Instead, it just handles things like note starting and stopping times, dynamics and tempos. Alongside much smaller file sizes, you also don't have to process stuff at 44.1 kHz anymore, you only need to process stuff at whatever your smallest note duration is saving immense amount of computer power. At least immense amounts back in the 80s-90s. Nowadays it's nothing to run some 44.1 kHz audio alongside your game play.
The MIDI 'music' you're nostalgic about is really just the sounds from the wave tables they used in those days. We don't use them much anymore because honestly, better stuff has come around in the meantime. Well, different stuff. Who's to say what's 'better'? I mean, analog synths are making a huge comeback these days. Interesting tidbit: Roland, responsible for some very quality analog synths way back in the day said (probably sometime in the 80s) that they were going completely digital and would never look back. Well, they released their first analog synth in decades a couple years back. Audio trends change all the time. Before you know it, those sounds you're nostalgic for will be fashionable again.
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