You could argue fair use or transformative work, but that would only come in to play after an initial DMCA takedown has been filed, and a counter-claim then filed in response.
It's possible, maybe even likely, that either of those has the potentially of being valid and winning the counter-claim, but even getting to that point as an individual vs a corporate law division is just not realistic due to the cost/time investment.
None of that of course accounts for what Twitch is going to do in this situation either. Are they or will they start issuing warnings/bans to streamers for getting too many DMCAs? What happens if an algo crunches through all 20k clips some streamer has saved and fires off 500 DMCAs against them in a day? Streamers are going to be wise to just purge everything and stay away from music entirely because the risk is too great to their livelihood until Twitch comes up with some kind of official stance/response (lol like that's happening this year).
Edit: it's entirely likely, or even probable, that videos like the above have been DMCA'd countless times, and they may have even won counter-claims. Youtube has a lot of automated systems in place for this already and has for years, while Twitch has been basically burying their head in the sand.