The game is launched by default using the "supporttool.exe". The servers are down causing it to error out and not have discord properly detect the game launch. Simply manually add the game to your games list on Discord to fix this and have it show. PROCESS:
1. In Discord open your User Settings
by clicking the gear icon at the bottom left.
2. Click Games
in the menu on the left hand side.
3. Click Add it!
at the top of the Game Activity page next to where it says "Not seeing your game?".
4. Select (2 WT)
from the dropdown list.
5. Click the title (2 WT) in the green box at the top to edit the name.
6. Rename it to Divinity: Original Sin II
If you have any issues I have attached a couple pictures.
This is not a fix. First off, Discord's auto-detection is based on the Executable's Path. For that portion, Discord has to update its list of recognized games to include Divinity. That said, they probably won't do so, given Divinity uses Gamebridge, the aforementioned feature.
Gamebridge is a Closed Beta SDK that Divinity uses to communicate with your Discord Client, initiating something called Rich Presence. It is through this that you can invite people to your Divinity session through Discord. I have it confirmed by Dabbit, the Head of Quality Assurance at Discord, that the problem is in Divinity. They have sent Lorian Studios a message about the issue, which only exists in the GOG edition, and so we must wait for Lorian Studios to patch it.
For confirmation, you can DM `Dabbit Prime#0896` via Discord, whom is in the Discord Testers guild