By exchanging and correlating data presumably? For example, anything I send or receive on Discord, I see reflected in my YouTube recommendations shortly after. It's downright egregious at times.

Most likely it's just run of the mill Google analytics/adsense tags in discord. Don't forget that discord is web tech and loads all kinds of JS bundles – including trackers. The best solution is to stop using discord, but the second best solution is to only use the web app version of Discord. When you use the web app, you can install adblock and anti-tracking extensions. The amount of data that Discord sends which gets blocked by these extensions is eye opening.