In fact, I reinstalled my pi-hole from scratch a couple of days ago, but on a different device which also works as a DNS server in my home network. I am not absolutely sure but I do not think that this has something to do with this issue.
Not yet done. Note: Restarting the router (Fritz!Box) leads to another unsolved issue of a different binding.
No update so far (Fritz!Box 7590).
No recent update. I have double checked the Kodi preparation. Actually, the Amazon FireTV got a firmware update today but the Kodi settings are still okay.
Both Yatse and Kore do work as expected.
Thank you for your suggestions, Ole @OMR!