@elijah286 The docs have been updated in the meantime:
The integration support multiple Spotify accounts at once. You don’t need to create another Spotify application in the Spotify Developer Portal and no modification to the configuration.yaml file is needed. Multiple Spotify accounts can be linked to a single Spotify application.
To add an additional Spotify account to Home Assistant, repeat only the steps in the Activating the Spotify integration section. However, make sure you go to the Spotify website and log out before you do so.