You can rename the session in the Session Manager:
Menu → Sessions → Manage Sessions → Select a session → Click rename.
BTW sessions are stored in $HOME/.local/share/kate/sessions. If you installed via flatpak, then it is probably in $HOME/.var/app/org.kde.kate/config/sessions
Do you have to reboot the whole computer, or is it enough to restart Kate only? Before you start Kate are the files itself are coming back? (In this case it is not a Kate-related issue)