Thanks you for your input.
I already anticipated that someone will tell me to wipe and install again. You are right with the assumption that I performed a dirty flash from Android 12 to 13 without installing Stock Android 13 on that device. I did that when upgrading from Android 11 to 12 and had not problems of any kind. But with all the backup processes come in place when wiping the phone I was to comfortable to do it this time. So it may be my fault.
@piero Thank you, too. Maybe that is something I will try to get rid off the misbehavior. Otherwise I will wait for 1.21 and hope for the best.
Is it possible to use the Error report (from developer options) menu tab to get a hint where the issue could come from or even identify the root cause of it? Browsing the logs after a crash would fit me more than re-configuring my phone including format+wipe in /e/OS recovery.