We are running into issues with our Desktop Office clients displaying auto-labels on the status bar. It is only on the Desktop Client.
We have an auto-label policy being applied to a SharePoint test site. The files are labeled. If you open that document in the web browser version of Word, it will display the label name on the bottom status bar. However, opening the document in the full O365 Word app will never display the label.
It's only for the auto-labels. If you apply a manual label to a document, it will display it both in the web and desktop client.
We have a demo tenant as well and the behavior happens there as well.