I never quite understood what AHI is .... is it an aditional hardware or soundcard?
It is a hardware abstraction layer for audio on the Amiga, much like Picasso96/Cybergraphics (collectively RTG) are for graphics.
The latter hook into the existing graphics.library to make sure that OS friendly apps that don't have unusual display requirements (read HAM etc) can open displays on the graphics card and render their interfaces normally.
Unfortunately, there was no similar audio.library for amigaos. OS friendly apps would allocate sound channels via audio.device and either play a sound then free up immediately or hold onto them until they are closed. Therefore, there was nothing that AHI could actually hook into, so instead it provided a completely new API. There is a device interface for high level apps that don't need fine control over the audio and a library level interface for ones that do.
AHI was, according to the author in his own docs, never intended to provide anything more than a stop gap until the OS gained proper audio hardware abstraction, but in the end, it became that standard itself.