Customizing Control Strip Icon for Spotify

I noticed a Spotify icon that displays in the control strip when I'm listening to Spotify on my phone. Can I customize this to display when I'm listening to it through my computer as a source as well? I haven't seen anything in System Preferences or Spotify's preferences that refers to this icon.

Also, what tools would I need to customize the behavior of an icon? Right now clicking on it displays rev/play/next and a progress bar for the song, but it would be cool for it to display the name of the song and a like button.