The only Atari machine featuring a DSP (the same Motorola 56001 as in NeXT machines) was the Falcon, which was introduced in 1992 - four years after the introduction of the NeXT system (and one year before NeXT black hardware was discontinued). Less than 20,000 Falcons were produced [1], whereas around 50,000 NeXT machines were built.

AFAIK the only Amiga that would have come with a DSP in a default configuration is the never released A3000+ which used an AT&T DSP3210. Prototypes of the A3000+ existed in 1991.

The major contribution of the original Atari STs to music were the built-in MIDI ports, which made it easy and simple to connect digital synthesizers, keyboards, etc. The music capabilities of a regular ST were rather constrained due to its outdated AY-3-8910/YM2149 sound chip (the SID in a C64 was more capable). Only the STE introduced PCM audio.

I guess someone more knowledgable about Amigas than me can add some facts about its sound capabilities.

[1] https://stcarchiv.de/stc1993/10/interview-atari-bob-gleadow