App Store doesn't show Mountain Lion as installed
When you purchase Mountain Lion in the App Store, it will first download an application called Install Mac OS X Mountain Lion.app to your /Applications directory. When the download is finished you're automatically prompted with the following window:



After that Mountain Lion installs and reboots into your Mountain Lion install. The installer application will be automatically removed - because most users won't need it anymore.

If you open the App Store, you're offered you to re-download Mountain Lion (hence the button says "download"), which could be useful if you wanted to create an install USB with the .dmg image.

You can verify if you have installed Mountain Lion by going to the menubar and click: Apple icon &#x2192; About this Mac. In case you have Mountain Lion installed, you'll see this: