I'm not feeling particularly alarmist about discord's cloning of the Twitch Drops concept but I'm very skeptical it'll work, if I'm being honest.
Twitch Drops take advantage of a very specific ecosystem wherein mid-popularity streamers are constantly looking for ways to draw in people outside of their core audience, and twitch rewards those streamers for playing along by playing sponsored games by handing them algorithmic boosts and front page slots, and a dedicated search interface for people using the feature. Which is all to say, twitch provides tangible incentives for people who think that drops is a feature they want to use.
Streaming on discord to your friends is not a "discoverability" activity. By and large I stream to single people, small groups, or to my girlfriend's server. In those situations I don't generally want strangers joining. I usually just want to hang out and play games I'm already playing. In that context, discord's "quests" have very little to offer. I don't need discoverability and discord doesn't offer it anyway, so the incentivization here is basically limited to really wanting, to take an example from a recent stream my girlfriend did, a purple duck in hitman. Even assuming I was the kind of person who cared about that, the absolute best-case scenario here is that they get me to switch which game I'm playing for 30 mins or so in a chat where probably 2-3 other people are playing their own games and streaming them.
Is this effective marketing?? Maybe??? If you get lucky and the game is fun to watch for only 15 minutes, or if you successfully pull someone back into a game they'd lapsed on. But by and large the conditions that turn twitch drops streams into 3-4 hour marathons with wide audiences just aren't present. The social pressure is (intentionally) nonexistent. And thus, I think that among the people who don't simply ignore this feature, it's going to be viewed as a silly novelty at best.