It's a thing with bazzite and most of the Ublue projects, you can install Flatpaks globally, but they default to per user. It's designed to be a per-user experience as this is the core of Ublue so you would have say a work image and game image etc rather than the traditional big blob of pakages.

This should add the flathub remote to the system and then install all the existing user packages into the system level. Then removes all the user level packages.

flatpak --system remote-add --if-not-exists flathub https://flathub.org/repo/flathub.flatpakrepo
flatpak --system install $(flatpak list --columns=application)
flatpak --user remove $(flatpak list --columns=application)

Personally I would just pick all the ones you'd like to be global (system level) and leave the rest at the user level.

It's a flatpak (and snap and appimage) thing; you only install it for a user by default, not for the whole system.

Just making sure: Did you check if the second user actually doesn't have access to the installed apps? Cause that prompt appears everytime you log-in.