[schapplm]

Since a few days (weeks) I encounter a strange error: ...

The procedure of the addon for a database file on a network share is to copy it on the local filesystem (.kodi/userdata/addon_data/service.watchedlist/dbcopy or a similar path). Then the addon runs as normal and copies the file back to the network share.
So my guess is, that already the part of copying does not work for you. If you enable the debug logging (for kodi and watchedlist), your log file should contain something similar to this:

10:52:08.514 T:139943538456320  NOTICE: service.watchedlist: created directory /home/moritz/.kodi/userdata/addon_data/service.watchedlist/dbcopy
10:52:09.495 T:139943538456320  NOTICE: service.watchedlist: Dropbox database downloaded: /watchedlist.db -> /home/moritz/.kodi/userdata/addon_data/service.watchedlist/dropbox.db
...
10:52:09.499 T:139943538456320    INFO: service.watchedlist: database_backup: database backup copy created to /home/moritz/.kodi/userdata/addon_data/service.watchedlist/dbcopy/20171228_105209-watchedlist.db.zip
10:52:09.499 T:139943538456320 WARNING: FileNFS::...
10:52:09.503 T:139943538456320   DEBUG: CNFSFile::...
10:52:09.509 T:139943538456320   DEBUG: service.watchedlist: copied db file /home/moritz/.kodi/userdata/addon_data/service.watchedlist/dbcopy/watchedlist.db -> nfs://192.168.2.21/volume1/Archiv1/Anwendungsdaten/XBMC/service.watchedlist/watchedlist_nw.db. Success: 1
10:52:09.511 T:139943538456320  NOTICE: service.watchedlist: created directory /home/moritz/.kodi/userdata/addon_data/service.watchedlist/dbcopy

I used NFS but I also tested this with SMB network shares before. Perhaps you also have a corrupted database file already in your userdata folder.