To my knowledge, those systems that supported it, was due to the hardware, not software.
Hardware that did not support it, would have dedicated music channels for sound effects, or simply categorize sound as unimportant and make that being muted, for a sound effect and wait until the next note plays.
The OPL sound chip from Yamaha had the first one, where the C64 SID chip had the second one. Only 3 channels, and if the game designer wanted an SFX, one of the 3 channel's sound would stop, and not play until the next note came along. This sometimes created a gap of sound, whereas with the OPL chip, it had, say... 4 channel polyphony with 6 channel virtual sounds. It could only play 4 channels at the time, but would move sound to one of the spare channels so it could make room for sfx. Sounds would play in order of priority, where, if more sounds were asked to play than polyphony would allow, it would not play it, or move its sound to a virtual buffer so it could start playing the moment the polyphony went down.