Am I missing something or can my Amiga or MorphOS machines only have one program using AHI at a time?
It depends. AHI can be used in two modes: exclusive mode and shared mode. Exclusive mode uses "music unit" settings. It may be used only by one application, and all other applications are ruled out. This mode is usually used by applications requiring low audio latency and full realtime control. Shared mode on the other hand may be used by many applications at once. Number of sounds playing simultaneously is limited by AHI settings.
There is also a possibility to "cheat", which mean making exclusive mode not so exclusive. It requires AHI v6. In music unit settings one has to choose "Unit 0:" mode. Then exclusive mode is redirected into shared one, at a slight quality loss and responsivenes degradation (which may be important, or not, depending on application).