I just found the error and I want to share it here:
I had two Domains: mydomain.de and mydomain.name. The name-domain was my testdomain for the original setup of MIAB. After all worked, I moved the production domain (de-domain) also to MIAB.
The mail-subdomain was first mail.mydomain.name and later mail.mydomain.de. But in the SQLite DB was still the path for the name-domain. The DB is in /home/user-data/mail/roundcube and I copied it to my computer, opened it with DB Browser for SQLite and look in the table carddav_addressbooks. In the column url was the path https://mail.mydomain.name:firstname.lastname@example.org/contacts/
I just changed the TLD from name to de and copied the file back to the server and and worked, but don’t forget to set the file rights:
chown -R www-data:www-data roundcube.sqlite