[T]he App Store recognizes that you already own the app, but it doesn’t treat it as if you bought it from the App Store. Meaning, on the App’s page within the App Store you’ll see “Installed” instead of the purchase price. However, you won’t see it in your list of apps underneath the “Purchases” tab and you won’t get updates for that app via the Mac App Store.

The Mac App Store won’t let you buy apps that are already “Installed”. According to Cabel, if you want to purchase an app through the Mac App Store that you already own, you have to drag your current app into the trash.