On first launch the game just shows a black screen. As other reports noted this is related to the game not running properly in DX10 mode.

User kaktuspalme has provided a solution in the relevant Proton github issue 641 thread here. First, launch the game once. This makes sure the Proton instance is created. Then copy two DX10 dll files from base Anno 1404 Proton instance to the Venice instance:

cp ~/.steam/steam/steamapps/compatdata/33250/pfx/drive_c/windows/syswow64/d3dcompiler_40.dll ~/.steam/steam/steamapps/compatdata/33350/pfx/drive_c/windows/syswow64/
cp ~/.steam/steam/steamapps/compatdata/33250/pfx/drive_c/windows/syswow64/d3dx10_40.dll ~/.steam/steam/steamapps/compatdata/33350/pfx/drive_c/windows/syswow64/

If .steam is not located in your home directory replace ~ with the correct path.

After the tinkering steps the game runs flawlessly in Proton 9.0-2.