There are two LR licensing models each with their own installer and program, one with a serial number, and one with a CC login.
You apparently have the serial-number version installed but you purchased a license for the CC-login one.
What you should do is uninstall the LR5 you have and reboot, then install LR from the CC Manager desktop app—the icon that looks like a lazy-8 or infinity symbol but is really two C’s facing each other. That’s the one that you’re paying $10/month for, not the serial-number one.
If you don’t yet have the CC Desktop app, then go to http://www.adobe.com, login with your Adobe ID that you used to purchase your CC license, and click the APPS / Desktop icon then click on the Download button under the LR icon..
It says something about installing the Creative Cloud Desktop App. Unfortunately I'm running on Ubuntu under Wine. I chose Lightroom over the completiton because it works under WINE. The Creative Cloud installer, however, does not look like it will install on linux (according to the interwebs). Is there any way to install Lightroom without having to install the Creative Cloud desktop app? Or can I purchase just a serial number for the application?
You can still buy a serial number for the serial-number-licensed LR using the Buy button on the Adobe Products page:
Be sure you're purchasing the full version for $149 not the upgrade version that will require a previous version's serial number, unless you actually have had an older version of LR before.
You will probably also want to cancel your CC PP subscription so you're not paying for something you cannot use. I think there will be a cancel link in your Plans list after you login to Adobe.com and click into Manage account, but eventually you'll have to contact Adobe, either via chat or phone to finalize the cancellation. If it is within the first month maybe there isn't a penalty, but I'm not sure.