cmcjacob can you explain how you integrated the shared authentication ?
The flarum_session
cookie contains the Session ID of Flarum, which is a standard PHP session. It always exists, whether you are logged in or not.
To log the user in from another app and immediately have them logged in inside Flarum with that cookie, you'd have to boot the session from your own app, set the session ID in the same cookie as Flarum then set the necessary attributes in the session to indicate which user is logged in. This requires having both apps on the same server for PHP to be able to share the session.
Another option would be to set your other app as an auth provider in Flarum (see the Passport login extension for example), or redirect users to a signed url on the forum that would take care of immediately logging them in with the user ID provided.
If I understand correctly you have only performed an API call to login, which returns an auth token. That token can be used to make API calls for the user, or must be set inside of the visitor PHP session if you want to have the user logged in on the web interface of Flarum.