Could solve the problem. It was an issue from upgrading to MIUI 8.3. You have to do a full wipe of the phone.
Here the steps to solve the problem:

Backup the data with Option “Backup & Restore” in the MIUI-Settings (under Additional Settings) Backup with TWRP Backup all your data (Music, Pictures, Videos…) - they won’t be in the Xiaomi or TWRP Backup Do a full wipe of system, data… in TWRP Install the latest weekly build from xiaomi.eu Restore from the Xiaomi-Backup Install DavDroid - and it will work. It even fixed some other issues i had from the upgrades in MIUI

Of course it will be a bit work, but it’s worth anyway. So always make a Backup and a Restore from the Xiaomi-App from time to time…

@rfc2822 And the MIUI issue is already mentioned in the FAQs I believe…

Thanks! Now it works. Keep up the great work!

@TheFreeBeginner Thanks for confirming the solution.

Thanks to a very nice person we met at Nextcloud conf 2017 we were able to personally take some important screenshots from a Xiaomi device and managed to find out what you need to do to get DAVdroid working with Xiaomi devices. Please have a look at these screenshots and on the colorized sections (stuff that belongs together is colored (orange, blue and pink) the same way).

Don’t forget to also remove your battery restrictions (if you want DAVdroid to sync automatically):

Thanks for the explanation.