Permanent icon on all Android screens
I am also only an F-Secure user. Sorry I am not really know about Android platform - but it sounds that Pixel 6 with Android 12. Is it right?
If so, I cannot check this situation with my own devices. But based on Android 11 (and Android 9) - there is also F-Secure SAFE notification logo. There is always, at least, one. Description with something like "F-Secure SAFAE should be ON/active to provide protection for you/device. And, so, they should show this 'notification' as a prevention to being closed/killed by Android" (sorry for rough meanings).
Perhaps, on your screenshot - this logo is something like described by me (but for Android 12 design).
Additionally, can you tap it (logo) or something like that? And check if it will be open any option or guide about.