**This issue is now resolved, in build 16.0.7369.2054.
Managed metadata SharePoint document properties can not be edited in the rich client Office applications starting in build 16.0.7167.2040, this is a defect and not by design.
Screen shot of working (prior to 7167)
Not working (7167+)
Roll back the Office 2016 client to a version prior to 7167 or edit the properties with the SharePoint UI.
The product group is actively working on this issue right now, check back to this blog post for updates.