Only user-wide installation is done when installing Flatpak apps in AppCenter, with normal or admin user. System-wide installation can be done in Terminal. But then it won't be reflected in AppCenter. Which means no update notification for those system-wide installed apps.
Is it possible to do system-wide installation in AppCenter?
It is not currently possible.
However, there is ongoing work to support this. I can't give a specific date as to when that feature would be ready, but I would hope it would be complete by the release of elementary OS 6.0
It is possible to manage system-wide Flatpaks from within AppCenter. You have to add a system remote to make this functionality available. To add Flathub as a system remote you would use the following command in Terminal.
$ sudo flatpak remote-add --if-not-exists --system flathub \
https://flathub.org/repo/flathub.flatpakrepo
Once a system remote has been configured, Flatpaks can be installed system-wide directly from AppCenter. Just note that it might require a restart for the option to appear.
Developers of the Elementary OS are not answering questions on forum. Report it on Elementary OS github. Maybe they will answer. But I doubt. Besides I recommend to read the guide: flatpak guide
