My thought on this: people hate making new accounts.
There's like a threshold in the human brain where the desire for membership has to exceed the loathing human beings have for account creation. Like if Mastodon didn't require users to make up their own passwords, I bet their userbase would double overnight. It's the reason why so many apps now just text you a one-time pass, so you don't have to remember your own.
What Discord does is lure people in with invites from your existing irl social network, your guild, your subreddit, whatever. And once they get you over the password creation hump you're now part of a walled garden ecosystem with thousands of chatrooms that can be accessed easily if you just have the link. Discord increases engagement because it allows users to be lazy. Anyone who makes a server there taps into that massive snowballing userbase.
It is really funny, though, how the game store didn't work out. The paradigm seems to fall apart a bit once credit cards get involved. I imagine people don't want their financial info tied to the account they use to trade porn.