Solution link (https://www.reddit.com/r/SteamDeck/comments/wu226c/ubisoft_connect_is_installed_and_running_but/).

I discovered a solution and was able to play it on my Steam Deck successfully:

Install Ubisoft Connect as a Non-Steam Game.

Download the installer with a browser in Desktop Mode

Add the installer as a Non-Steam Game and set Compatibility to the latest Proton-Version

Run it and finish the installation

Change the path to the exe in the properties if the Non-Steam Game to the UbisoftConnect.exe at \home.local\share\steam\steamapps\compatdata<long-number>\pfx\drive_c<install-location-you-choose-before>

Run the Non-Steam Game again and login to Ubisoft

Quit it.

Install the Ubisoft-Game you bought on Steam directly from Steam

Try to start it once to get the Error-Message, that Ubisoft Client is not found.

Copy all files from \home.local\share\steam\steamapps\compatdata<long-number-of-Ubisoft-Connect-Installation>\pfx

to

\home.local\share\steam\steamapps\compatdata<Steam-ID-of-the-Ubsisoft-Game>\pfx overwriting all existing files.

Try to launch the Game again through Steam and it should open Ubisoft Connect and start the game without errors.