First off, make sure your new computer is able to run Photoshop Elements 2019. You don't say whether it is a Windows PC or an Apple Mac. If it is a brand-new Mac with macOS 11 Big Sur, then you will need to upgrade to Photoshop Elements 2021 as macOS 11 Big Sur does not support 32-bit applications.
So assuming it's a nice, new, shiny Windows 10 PC 🙂 :
If you're not going to use Photoshop Elements 2019 on your old computer, then first sign out of Photoshop Elements on that computer. That will ensure you still have 2 activations left on your account.
Then on your new computer, sign in to your Adobe account at www.adobe.com. Under Plans > Order history you should see your original Photoshop Elements purchase. You can retrieve your original serial number from there, and also download Photoshop Elements 2019. Then just install Photoshop Elements on your new computer and enter your serial number when requested.
If you use the Photoshop Elements 2019 Organizer and want to transfer your catalog and all the images to your new computer, then before you do all the above, on your old computer make a full backup, and save it on a removable drive. Then after you have completed the installation as described above on your new computer, connect the removable drive to your new computer and restore the backup file.
Here's the relevant section in the User Guide for backing up and restoring catalogs: