Why no name is presented on DECT Handset, even if MobileDock is operated on a fully compatible Gigaset DECT System and a matching name is available in the smartphone contacts?

For presentation of contact names, MobileDock requires access rights to the smartphone contacts.
Android: Activate 'Contact sharing' in Settings - Bluetooth - Settings for MobileDock
iOS: Activate 'Sync Contacts' in Settings - Bluetooth - Settings for MobileDock