Can launching an app trigger the launch of another app?
Open Automator, and choose 'Application' from the pane that appears.

In the sidebar, there should be an item called 'Utilities'. Click this, and drag two 'Launch Application' actions into the main workflow pane. In the first one, choose the application you want to use with Spotify, and in the second one, choose Spotify. (See the picture below.)



Then, save the document with a name you'll remember (something like 'Spotify Launcher') in /Applications, and replace Spotify in your dock with the Spotify Launcher app that's been created. It will launch both Spotify and the companion app you want to use with Spotify whenever you invoke the application.

Hope that helps!