Thanks for all the great suggestions, here an update:
@DNSB: Tried that already before, even deleted all Calibre related folder under Users/<myuser>/AppData/Local. Same behavior. I agree with your second post, I have forgotten to verify the network settings in safe mode.
@PeterT: Installed a portable version on my Windows computer and linked to a local directory. Great surprise - it worked!
Re-linked to my database on Dropbox - and it's still working - which are great news! There is nothing interfering on my Dropbox account, because my Calibre Apple version is using the same database (and it's always either/or).
@davidfor: Have not tried this, but the portable version of Calibre is working, so I don't think anymore this is a network (or AV/firewall) related issue. And I had the same behavior with Kaspersky switched off.
I will next try to find out the differences between the installed version of Calibre and the portable version. I'm still wondering why the normally installed version stopped working on Windows (but not on macOS), with Calibre versions in-sync.
Thanks again for all your help, suggestions and support!
Claus