Premiere Elements program "registration" is not the same as "activation". The Adobe document on reasons for registering an Adobe product can be found HERE. With many Premiere Elements versions, registration is done automatically as part of the program's installation/activation. Once the program is registered, the user is expected to find a listing for the registered program under Plans & Products/Products/View Your Products in the user's Adobe account which is accessed at the Adobe web site.
If for some reason registration is not done at installation/activation, the program can still be run in the absence of registration. And, if registration is wanted after program installation/activation, the opened project offers a route to registration via its Help Menu. See*
Premiere Elements 4, 7, 8.0/8.0.1
Premiere Elements 9.0/9.0.1, 10, 11, 12.0/12.1
Help Menu/"Complete/Update Adobe ID Profile"
Premiere Elements 13.0/13.1, 14.0/14.1, 15
Help Menu/"Manage My Account"
In the case of Premiere Elements 4, "Registration" was found active in its Help Menu, but its use gave the error message "Registration - Connection Failure". In the cases of Premiere Elements 7 and 8.0/8.0.1, "Registration" was grayed out in the Help Menu. It was decided to offer a path to Premiere Elements registration at the Adobe web site. Premiere Elements 7 was used for the test case.
The following should work for all versions of Premiere Elements; however, the only test case here was using Premiere Elements 7.
1. Go to your Adobe account at the Adobe web site http://www.adobe.com and sign in with your Adobe ID and Password (See top right of the Adobe web site page that opens.)
2. After entering your Adobe ID and Password, a page should open with the headers of Overview, Plans & Products, Account, Security & Privacy, Communication, and Profile. Select Plans & Products.
3. In the Plans & Products section, go to the right side of the page to Products/View Your Products, and select View Your Products.
4. At the bottom of the list of your already registered products, there should be a "Register an older product" section which asks for the serial number of your product that you want to register. After the serial number has been entered in the space provided, press the Enter key on the computer's keyboard. The result should be the name of your product (in this case Adobe Premiere Elements 7) now appearing in the list of your registered products.
(Side note: Adobe Photoshop Elements 7 could also be registered here using its specific serial number.)
Also of interest may be the Adobe document describing transferring license of Adobe product. See HERE.
*These Help Menu/options (if they are found in a working state) should be opening to the Adobe web site page cited in SOLUTION.
**Study done using Premiere Elements on Windows 7 Professional SP1 64 bit. Also applicable to Windows 10 Home 64 bit.