The desktop entry needs a "%U" at the end of the exec line to open links outside the browser. This issue is related but seems to be resolved: https://github.com/zen-browser/desktop/issues/161
For the icon to work in some cases it also needs "StartupWMClass=zen-alpha". It doesn't seem to use the file on GitHub from the same release. Once I installed the Flatpak it also uses it's patterned icon for the AUR package.
Pinned Comments
Larvey commented on 2024-12-24 02:33 (UTC)
As of Beta build 1.0.2-b.4, zen no longer provides an avx2 optimized build. If you are using this package (zen-browser-bin), there is nothing you need to do, but if you are using zen-browser-avx2-bin, then it is recommended you install this package, and select yes when asked if you'd like to remove the conflicting avx2 package. A merge request has also been put in for the avx2 package.
More information from the Zen developers can be found here: https://github.com/zen-browser/desktop/wiki/Why-have-optimized-builds-been-removed%3F