I have installed Photoshop Elements on a new computer and signed out from the old computer. However the new installation is only allowing me to use Elements for a 30 day trial, depite my Adobe Creative Cloud running as well. Shouldn't it automatically register my version?
If you are seeing a dialog something like the following, click on the License This Software button and you will be taken to a screen to enter your license number.
As John says, Photoshop Elements is not part of the Creative Cloud. You must buy a separate license for it. (But because Elements 2021 now allows you to save your files to the Adobe Cloud, the CC application will be installed on your computer, even if you only purchased Elements.)