Is this supposed to be a distributed DB that auto-synchronizes instances? Documentation doesn't seem to say anything about that.

If not, it seems like it would be quite a bit of work to implement the synchronization... and I don't understand why one would use it otherwise.

This documentation page[1] seems pretty clear. One primary at a time, any number of read replicas that automatically proxy writes to the primary, when compute scales to zero the data is in object storage and a new primary can spin up elsewhere.

[1]: https://docs.bunny.net/database/replication