Hi Torsten,
When we use Outlook desktop or Outlook for Mac client for our email account, the profile could be corrupted by some probable causes which leads to some issues like features not work or any buttons are missing. In this situation, the general steps we can try are to recreate a new profile/identity.
To my knowledge, there are no such settings in Outlook Web App that can control the Outlook for Mac features like saving an attachment. In this situation, to confirm if the issue is related to Office 365 Online mailbox settings, I would suggest you create a new profile/identity and set up a Hotmail or Gmail email account to see the issue persists.
Moreover, I have provided you an Office 365 test email account via Private Message. You can also set up the test account our your Outlook for Mac for test. To access the private message, please:
1. Go to the Your details section on the right side of the community site.
2. Click Private Messages.
3. Click the subject title of the response to read the message.
Please let us know the results after your tests. Awaiting your updates.
Best Regards,
Alex