I did try that, when swapping contents of /usr/share/gnome-shell/ between Ubuntu and Zorin. I'm also not working with any Zorin specific extensions for GNOME, only Ubuntu and user-installed ones, although as I detailed that distinction matters little in this case. To elaborate on that, the description for Ubuntu Dock reads
A dock for the gnome shell, default ubuntu experience. this extension is a modified version of dash to dock with different defaults. dash to dock can be installed to replace it and give more (but unsupported) configuration options.
I've tried both Dash to Dock and Ubuntu Dock with no results. The only sensible conclusions I can reach at the moment are: Zorin 16 ships with GNOME 3.38.4 and Ubuntu 21.04 with GNOME 3.38.5, so this minor release could be the culprit, although no changelogs seem to support this, or Zorin 16 is having trouble displaying these badges for whatever reason.
My suspicion is towards the latter.