The Flatpak version of Brave doesn’t support some extensions integrating with their desktop counterparts (like password managers, Gnome Shell Integration, etc.) due the natural limitation that exists with the sandbox provided by Flatpak.
However, after some discussion on the flatpak/xdg-desktop-portal repository (see issue #705 and issue #1537) it seems the solution to enable this feature is on the flatpak/xdg-native-messaging-proxy repository (see the mail thread listed at the end of issue #1537).
So the feature request here is to integrate with flatpak/xdg-desktop-portal