Play Store icons and downloaded app icons, same or different
Icons are also provided by your icon theme. But for convenience both kept nearly similar. 
Yes, it is possible to keep two different icons. But, that will create confusion to your user. I can cite you an example of an app. This app is SuperSu. It can change up to 7 icons ( including 1 invisible ) if a user opts.
You can also change app icons by using some app that replaces icon from the app and from your system. For an example see this app.
You can also follow this tutorial to change the icon.