Not familiar with monthly subscription, but seems like with all products it needs a serial number for activation.
So I would call customer support and expalin you can find no serial number. Have your order number and prooof of purchase handy.
I could finally fix the problem with the help of Adobe support. For everyone with the same problem:
1. Deinstall Photoshop.
2. Delete the following files (replace "Programme" with the Program Files-Folder of your language):
C:\Programme (x86)\Common Files\Adobe\caps\"caps.db"
C:\Programme (x86)\Common Files\Adobe\backup\"caps.db"
C:\Programme (x86)\Common Files\Adobe\caps\"media_db.db"
C:\Programme (x86)\Common Files\Adobe\Adobe PCD\"pcd.db"
C:\Programme (x86)\Common Files\Adobe\Adobe PCD\cache\"cache.db"
3. Redownload and reinstall Photoshop.
4. Start Photoshop and login with your Adobe-ID.