I had the trial version installed for some days and than bought it from the AppStore. It was downloaded, than I confirm the question to to close the program because it was running. Than another message said that the installation failed and I should try again from the list of my bought apps. I tried that but the installation stopped (never continuing progress bar on grayed out Cookie app symbol in the Launcher). I reseted my MacBook Pro and installed the app again from the AppStore (this time I've manually closed the running Cookie app before). This time the installation succeeded but I'm not too sure if the program is really licensed now. There is no message or something in the help menu nor everywhere else saying that the program currently is registered or bought from the AppStore.
The AppStore version doesn't specifically say anywhere that it is licensed. The easiest way to check if you have the App Store version installed is to open Preferences and see if you have a tab named "License". if there is NO "License" tab, then you have the App store version installed, and don't need to do anything else.
If you see the License tab, this means you have the trial version installed, and should quit/delete and then re-download from the App Store.