tl;dr: If you want to run GNOME Software on MATE or XFCE you need to set an environment variable like
Long version: In the software application we have the problem where applications have the same name and summary, but are targeted against different desktops. We know when an app targets a specific desktop from the AppStream metadata (which currently uses a heuristic from the .desktop file) so we could filter these out client side. At the moment searching for notes gives you two similarly looking results results provided by two different applications: bijiben (GNOME) and xfce4-notes (XFCE). Also, because of the shared history, a lot of the MATE applications have the same name as the GNOME ones. This makes bad UI.
The default is
GNOME_SOFTWARE_COMPATIBLE_PROJECTS=GNOME (well, it is gnome-software after-all…) but if you want to show MATE projects and hide the core GNOME ones, just use a setting of
MATE. If you want XFCE, GNOME and MATE apps to show, just use
As usual, comments and suggestions welcome.