Had to replace the out-dated UPlay with UbisoftConnect. Steps to reproduce:

1. Vanilla installation of the game.
2. BEFORE launching the game for the first time,
3. Download Ubisoft Connect from Ubisoft's website: ubisoftconnect.com
4. Find your game installation folder. Something akin to ~/.local/share/Steam/steamapps/common/Assassin's Creed Unity
5. Under that folder, go to Support/Software/GameLauncher
6. Remove the Uplay file you find here and replace it with the UbisoftConnectInstaller.exe that you downloaded.
7. In the top-level game installation folder, find a file that end in .vdf. "find . -name *.vdf" should return this filename.
8. Open that file in your editor of choice and find a line towards the end that looks like "process 1" "%INSTALLDIR%\\Support\\Software\\GameLauncher\\UplayInstaller.exe" and change "UplayInstaller.exe" such that it points at the UbisoftConnectInstaller.exe file you just placed.
9. Launch the game for the first time
10. This should go through the usual DX11 steps as well as installing Ubisoft Connect. It may have you sign in to your Ubisoft Connect account, but it will add AC Unity to your Ubisoft library.

This should "just work" after that. There may be other settings you can add to improve stability or graphics, but this made running the game feel very vanilla. As if this is how it is meant to work. The one caveat is that exiting the game does not stop the game. You may have to stop it in Steam as well. This process was verified with AC Syndicate as well.