There's a fairly common method in most streaming software now to split out the audio into different feeds so that the music is streamed but not saved into VODs.
It's something that streamers themselves have to do, and most don't. If the streamer doesn't separate out the audio from their mic vs the background music, then there is nothing that Twitch or anybody else can do. Separating two audio tracks that have been flattened into one (and then compressed to hell) is like trying to unmix paint.