It might be that Discord is remembering user information in iCloud in a way that persists following reinstall, similar to how some games save progress. Clearing the password entry would thus be insufficient.

I have iCloud disabled, so that’s not it.

This is because each app has their own private keychain (separate from the saved passwords) which is not cleared when the app is deleted. Apple tried to change the behavior and have it wiped on delete, but many developers complained. This allows apps to track you across installs of the app.

That's...actually terrible. A bit of a search around seems to say that the only way to actually wipe it properly is the do a full device reset, which is obviously completely unacceptable.

Yeesh :/