I found out something else interesting:
Since the new installation, the web search works on my account "joerg@XXXXX-XXXXX.ch") (This did not work during the first installation, although I had imported the data in the same way).
The web search does not work on the shared account "familie@XXXXX-XXXXX.ch".
However, the web search of the shared account "familie" works when I search in the inbox "Favoriten".
Work:
Doesn't work: