Did you purchase a LR 6, the upgrade is about US $80, the non-upgrade about $150, or did you purchase an annual CC Photography subscription that’s US $10/month?
Have you looked at your product & plans list assigned to your AdobeID after logging in to adobe.com and click on manage-account?
I purchased the annual CC Photography subscription for 10$ a month, when I go to my adobe and look under the products and plans list is doesn't give me any options except to see what plan I have.
If you have the CC Photography Plan then you won't have a serial number, you merely need to login with the AdobeID associated with your CC Photography Plan. Help / Sign In within LR is one way to do this.
You say you have the options to see what plan you have, but you don't say if you actually do see the plan listed or not. Do you see a CC Photography Plan associated with your AdobeID?
If so, if you do have a CC plan associated with your AdobeID, then follow these troubleshooting steps to relicense LR:
If the issue is that you don't have a CC plan associated with your AdobeID then perhaps you have multiple IDs and you should log in with another one. You can also try chatting with Adobe directly as they can view their server information directly.