Hi @Markus Männl ,
Welcome to Q&A forum!
Considering that there is no problem with OneDrive and Teams, I would suggest that you try to reset Microsoft 365 Apps activation state to check this issue. For more inforamtion, please refer to: Reset Microsoft 365 Apps for enterprise activation state.
If the above does not work, please refer to the steps in the following official KB to check again.
Unlicensed Product and activation errors in Office
Hope the information is helpful to you. Any updates, pelase let me know.
If the answer is helpful, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.