Flash is only required when subscribed to Google Play Music All Access / Youtube Red, otherwise a free account can readily enable HTML5 audio in the options, which works in Chrome/Safari/Firefox (just tested again to be sure) and probably Edge also.

Google Play Music uses MSE (Media Source Extensions) to stream MP3. Firefox doesn't support MP3 in MSE streams. Google Play Music could use AAC or Opus in MSE, like YouTube does, but they prefer MP3 because some Android devices have MP3 hardware decoders that (may!) use less battery power than software decoders.

YouTube encodes all their content with in multiple formats: AAC and Opus, H.264 and VP9. I don't know why Google Play Music can't encode their music streams in both MP3 and Opus. They would save on bandwidth by using Opus.

Here is the Firefox bug to add support for MP3 in MSE: https://bugzilla.mozilla.org/show_bug.cgi?id=1169485

Hmm it works on chromium without flash. I wonder what chromium is doing that firefox isn't doing.

EDIT: Tested this out on opera and it works fine. Does it have anything to do with blink?

which is a bit surprising considering how quick a company the size of Google could whip up a more modern approach to this!