Hi @jaquesv thanks for posting and welcome. There is no “registration” as such - PopClip is only available to buy on the Mac App Store and so if you have decided to buy PopClip, all you have to do is go to the Mac App Store, sign in with your Apple ID, and click the Buy button in the store.
The App Store’s installation process will automatically replace the trial version of the app with the full unlocked App Store version.
Your extensions will not be lost, because as @gkruijer mentions all your extensions and settings are stored in ~/Library and these are not touched by the Mac App Store when it installs the app.
(If you want to, you can delete PopClip.app from /Applications first before installing from the Mac App Store, but it is not necessary. However do not use an “uninstaller/cleaner” app to delete the trial, as this may well delete your extensions and settings which is exactly what you don’t want.)
In short, just click buy on App Store. There are no other steps and you won’t have to recreate your snippets.