There's no term for music from the 16-bit console war era. The reason is that the music sounded very different due to the hardware capabilities and limits of each system.

There are many comparison videos between snes and genesis which can be used to compare audio.

From Castlevania, Simon's theme.

From Street Fighter 2, Blanka's theme.

If there is one common technique that separates this music from the 8-bit era, it's "sampling".

Some people call older game music Chiptunes. It doesn't necessarily fit all game music, but there is quite a lot of overlap if you do some research.

Searching YouTube for "chiptunes" will provide quite a few links that sounds like game music, even if they are not directly from games.

Shorter answer:

Personally, I use "Game" as the primary genre for game music in general and "Game Remix" for (I'll wager you guessed this) Remixes of old Beatles songs... heh. I tend to use multiple tags for genre, though, so I tag with console (e.g. "SNES", "NES", "Genesis", "Saturn", et cetera), as well.

"Chiptune" means 'tunes generated by embedded sound chips' in my circles. I've never seen anyone apply that kind of a bit standard to that, before, but there certainly are categorical differences.

• "Generated sound" or "waveform synthesis" music: 8-bit music is generally synthesized from a few basic component waveforms and mixed.
• "Tracked" or "Tracker" music: 16-bit generally runs a program like a Music Demo playing MODs. That is, it's synthesized on the fly from stored waveforms. Other names for this sort of thing are: Demo, MODs, instrumental synth, and so on. However, to have a meaningful sample, the sample must be recorded in some manner. Since there are multiple forms of doing that, (e.g. PCM, FM, etc) there are actually multiple terms that can apply. One gotcha is that sometimes you get waveform synthesis out of this era of music processor, too.
• "Streaming" music: 32+bit systems generally pipe in audio streams and don't synthesize anything except, sometimes, sound effects. Ironically, that means that in significant ways, modern gaming audio processing is less advanced than earlier systems. I find that odd.

Long and involved nerd answer:

Early sound chips generated their sounds by adjusting the frequency and amplitude of a few basic waveforms. These were generated by simple signal generation circuits and the chips that compiled these functions together are known as programmable sound generators (PSG). What I've seen pretty much always uses some combination of sine, (rarely) cosine, triangle, square, saw, and something noisey. (https://en.wikipedia.org/wiki/Sound_chip)

The home consoles were never MIDI. (fun fact: MIDI predates the NES by a year or two) A MIDI song is a set of commands for playing notes that are sent to various instruments. The recipient of those commands is supposed to play them. The results can be anything from terrible synthetic music to indistinguishable from a recording of a natural instrument or even an actual instrument. MIDI is a control language optimized for making music, not an audio format.

Since MS Windows 3.11's Multimedia extensions, it's been normal for MS to package in some sort of MIDI synthesizer. (which improved over time and then fell back to poor in either XP or 7. Not sure what support looks like in 8 and 10) This sets many opinions as to what MIDI can do, sadly.

Cheap MIDI synth. sounds cheap and be quite reminiscent of older styles of synthesis. (Some soundfonts exist specifically to narrow that gap even further.) High quality MIDI synth. can sound like a CD recording of an orchestra. If you think MIDI music has to suck, it's because you haven't heard MIDI used to direct good instrumentation and don't understand its potential.

http://www.iub.edu/~emusic/361/midi.htm

https://www.midi.org/

Amiga had a much more powerful sound chip than most systems of its era. (graphics, too, as it happens) One of the results of that was SoundTracker and the MOD format. Tracker music is a sequence of notes and effects on those notes generated from samples transformed and mixed. The notation part of the format of tracker music is actually quite like MIDI, but it carries along its desired samples and rules for using them. This makes it harder to upgrade but more likely to sound right.

https://en.wikipedia.org/wiki/Module_file

Related, but aside:

https://en.wikipedia.org/wiki/Audio_file_format

https://en.wikipedia.org/wiki/MOS_Technology_SID

The sound hardware of the times mentioned handled MIDI (and similar formats) differently.

Early devices did what I will call "synth midi" - using midi or some similar format to store notes and then a synthesizer chip would produce each tone.

Some years later, better hardware and more memory allowed "wavetable midi" which used real sampled sounds to produce the notes. It sounded more realistic, but still used the midi or similar format to store notes to be played. Similar technology with proprietary formats were used in other music software of the time like Scream Tracker et al, which stored the notes to be played and the samples to use in a single file.

I would call your 8-bit music "synth midi" and stuff that came later and used real sounds "wave-" or "wavetable midi". If there are official names for such genres I'd love to know.