This happens in both Tumbleweed & Garuda Linux. On tumbleweed, I installed Zoom via flatpak, AUR on Garuda; no errot or anything when I try to share audio, just doesn’t do anything, shares the window fine. I use KDE on X11. About a year ago, I successfully shared sound on Tumbleweed
You could try running the flatpak with a few flags.
--socket=pulseaudio is used to allow pulseaudio access but there is currently no
We can see here that the zoom flatpak is already built with that flag enabled.
After a little bit of digging I found this discussion on github
--filesystem=xdg-run/pipewire-0 is the way to enable pipewire
Try that perhaps.