Is there another account (preferably an Administrator) where you can login to run services.msc to see status of Duplicati service? Or Task Manager services tab can show service state, but can’t change it.
You can look around directly in Task Manager, e.g. what Duplicati* processes are on the Details tab?
Sysinternals Process Explorer is more powerful, and right-click → Properties → Image has Started which might shed light on when things started, Duplicati.GUI.TrayIcon.exe could start at logon, but the Duplicati Windows service functions (activated by Duplicati.WindowsService.exe) should start at boot.
Note using TrayIcon with Windows service needs some special setup, or it will run a second Duplicati, which confuses people. Make sure you see older backup job logs, with mystery gaps where logged off.
EDIT 1:
If you have no way to sneak in to see status without affecting status, another option is using log-file with –log-file-log-level=information or something. That should log the Duplicati startup, and any jobs that run.
Although I’m not sure how it could affect things, also test without a browser on Duplicati, and no scripts.
EDIT 2:
Is your backup destination something that logout could affect? What Duplicati version are you running?