I am wondering if it is possible to use just the Firebase authentication system together with your own API and database.
It has some nice features I want to take advantage of, however I do not wish to use their database or storage.
The application I am building is an Angular2 and express application with a MongoDB database.
Any answers will be greatly appreciated!
That definitely is possible: you can create your own identity provider that plugs into Firebase Authentication. This is often referred to as custom authentication. To implement this, you need a server (or other trusted process) where you authenticate your users and mint security tokens for then. You then pass this token to the user and have them pass it into Firebase.
But keep in mind: many Firebase features work fine without authentication. For example: if you use Cloud Messaging, Crash Reporting, Test Lab or many other features, you might not need to implement Firebase Authentication at all.