Hi AFRISINA-RAINER,
To help troubleshoot, let's have you take the following steps:
1) Ensure your iPhone is updated to iOS 16.0.2: Update your iPhone or iPad - Apple Support
2) Tap Settings > Focus
3) Turn ‘Share Across Devices’ to On if it is currently off.
4) Turn on one of the default Focus states, such as Do Not Disturb
5) Wait a few seconds, then turn that Focus state off
6) Check with the recipient to see if the message is still displayed.
If the issue continues, take those steps with each other device you own.
Reach out to Apple Support directly if the behavior persists: Contact Apple for support and service - Apple Support
Kind regards.