Macos – Disable unread count in the Skype dock icon

No.

The dock badging style is not changeable by the user, and also (I believe) not changeable by the application unless it extends the dock badge.

One option would be to keep the dock size small but put a large magnification on it so you can simply flick your mouse down to the Mail icon and it will magnify enough to read the badge.

Discord also doesn't show up for me in System Preferences -> Notifications, however there are two places where Discord notification settings are, both accessible from the app.

From the top ribbon, click Discord > Preferences and there's a Notifications menu. From there you can disable Desktop notifications.

Additionally, if you right click each channel you're in you can see the per-channel notification settings.

Finally, in order to actually clear the badge after I turned notifications off, I right clicked each channel and clicked "Mark as read".