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)
The answer might be how you open kate. I had the same issue and remembered that I set my keyboard shortcut to SUPER+t with the command kate --start “oldsessionnamehere”. This will automatically create an empty session with that name within .local/share/kate/sessions/
Thanks for the clue. I am using Linux Mint “Wilma” ver 22 . I installed kate via Flatpak. Now the session is stored in the folder $HOME/.var/app/org.kde.kate/data/kate/sessions