If you bought the application on the Mac App Store and it complains about being a demo version, it means that the one you’re using comes from this website or elsewhere instead. The ones from the Mac App Store don’t require any license key.
You might want to look in the Applications folder to see if you spot a second TabView/Tablatures (perphaps in a folder whose name is the one of the app).
Otherwise, to make sure that the App Store doesn’t get confused, you can simply delete the application, empty the trash and download it again. You won’t need to buy it again, of course: you’ll find it in your purchased apps section of the App Store.