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).

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

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Thanks for the explanation.