Kate is not installed as a flatpak. I think it came with the default snapshot,
as there's no Kate on flathub, only flathub-beta and kdeapps.

When I initialize or clone any git repository and open its files with Kate, the
Project pane on the left provided by the Projects plugin doesn't update to show
the contents of the repository.

Additionally, the Git pane, also provided by the Projects plugin, doesn't show
any changes if I edit and save any files in the repository, even when I click
the Refresh button.

This works fine with Kate installed in a distrobox container, as well as Kate
compiled from within a distrobox container.

Unsure if this is related, but the experimental Kate flatpak doesn't have a
functional Projects plugin either. I wonder if something is missing from both.
https://github.com/flathub/org.kde.kate/issues/32