Thanks for this hint.
Unfortunately I already tried this but it didn't help.
It appears to me that the installer via dietPi for piHole is maybe broken.
Anyway, I now have access to the web dashboard. What did the trick was:
- uninstall pihole & dependencies via dietPi-software
- fresh install and configuration via curl -sSL https://install.pi-hole.net | bash
All working now. We can close this topic.