"backend as a service"
Instant APIs
RESTful and GraphQL APIs are automatically generated based on the DB schemas.

roll out my auth (with Authjs for example) or use supabase auth?
Authjs has a Supabase adapter. (It uses the Supabase DB to store user and session data in a separate next_auth schema. It's a standalone Auth server that does not interface with Supabase Auth and therefore provides a different feature set)

Firebase

Firebase is more general, that includes 2 types of storage (databases), authentication, cloud functions, cloud storage buckets (for assets/media). and much more

firestore nosql DB.

firestore is free up to 10GB...
cloud firestore seems to be 1GB, 50k reads per day
hosting seems to be different from firestore cloud
firebase real time database is for publish/subscribe system designed for rapid ingestion and processing od data,
Cloud Firestore is designed for long-term data storage and retrieval.

Supabase

competition is open source alternative supabase. it is self hostable.
I think ppl prefer this alternative nowdays.