Hi there,
Adobe Photoshop Lightroom subscription asks for serial number under two conditions:
1. When you have just downloaded trial and it's 30 days validity is over Or you have reinstalled the trial on the same system.
2. Even after downloading the cloud subscription.
You will have to uninstall the trial version of lightroom from the sysstem and then login to creative.adobe.com and download the lightroom from there and sign with the adobe id and password, and it will no longer ask for serial number.
Thank you.