Lightroom via the Creative Cloud does not come with a license key. You have log on to your account through the Creative Cloud Application Manager and install from there.
If you are being asked for a license then it means you have the Perpetual Licensed version of Lightroom installed, you will have to first uninstall using your operating systems normal procedure. This will just remove the LR application and will not interfere with your Catalog, Preferences and other personal files.
Log on to your CC account and the application manager should show available for "install" proceed and everything should be fine.
Thanks for your reply. I installed via the CC application manager. It now shows that I'm running trial versions of LR and PS. Why should deleting a redoing the same thing again give a different result?
There are two licensing models for LR and each LR type has its own installer.
You would have trial versions of the CC-login-licensed LR and PS if you have not yet paid the $50/month for a full subscription or the $10/month for the Photography Plan subscription or at least if the payments have not been approved or the local CC Manager doesn’t know the payments have been approved.
You would have a trial version of LR that asked for a serial number if you installed LR from disc or downloaded an update from the Adobe Updates page: http://www.adobe.com/downloads/updates/ which only works to download non-CC products.
From what you’ve said, you installed PS and then maybe LR from the CC Manager, but then installed a LR update from the Adobe Downloads page, so have switched LR from the CC-licensed version to the serial-number-licensed version.
If you have actually paid for the Full-CC or Photography Plan subscription then you would be licensed via the CC-login, and merely need to uninstall the serial-number-licensed LR you accidentally installed, and sign out of the CC Manager desktop app, then perhaps reboot, and reinstall LR from the CC Manager app using the Install button next to LR in the Apps list.
Also, your message only included a filename not the actual contents of the screenshot.