All of the Firebase backend is proprietary, and their client code is open source.
The Supabase backend is here: https://github.com/supabase/supabase/tree/master/docker
This is the set up we use for every project on our hosted platform. These backend tools are either built by Supabase, supported by Supabase (we employ maintainers), or licensed with OSI-compliant licenses.
Our client libraries are all open source, Apache 2.0 licensed.
Supabase is open source in every definition of the word. I think the problem is our self-hosting docs, so we'll spend the day improving this.
Supabase builds on open source parts that everyone is already familiar with. Postgres for data store instead of firestore (which is a proprietary nosql Google cloud database with many quirks to keep you in). GoTrue for Auth which is an open source project by netlify. Real-time to provide Postgres changes over sockets open sourced by them. There are many parts which make up "supabase".
Supabase hosting provide a unified experience through their dashboard and manage all these parts for you. That isn't open source.
The actual UI kit for that is open source, though.
Dashboard doesn't stop you from migrating your backend infrastructure somewhere else. It does result in degraded developer experience which can be the biggest distinction.