Hi, I just installed Affinity Publisher 2 and am experiencing the same issue as you, @mahu. Any chance you figured out a fix?!
The first thing I tried was to restart my PC. I also tried clearing the icon cache (using instructions here: https://answers.microsoft.com/en-us/windows/forum/all/icons-broke-in-windows-search/d8f92c14-2ffe-43ec-87b0-c60fc16e2692); still no icon.
The icon for Publisher 2 shows in the task bar when I open the app, but it does NOT show when I search for the App in the Start menu or when I pin to Start. It pins as a blank box and is not clickable (does not open the app).
When I look in the Applications folder, it is clear that there I something amiss with the icon for V2
With V1, in this folder I can right click the icon and see click "Open file location", but there is no such option with V2 (I assume it has something to do with the new install method). Where do the version 2 apps get installed on Windows 10? Not sure what to do next to troubleshoot this.