We wrote a custom storage layer for postgres, so in our setup, postgres node (k8s pod actually) doesn't store any data, and it is easy to start/stop/reschedule it. So while we actually are DBaaS, the closest analogy here is Aurora or Alloy, not RDS-like setups.

A bit more details are on https://github.com/neondatabase/neon

Typically it means S3 and alike. Not being bounded by disk size of your server.

Or in their case they might refer to horizontal scalability of their storage layer which is independent from computing.

Yes, you will never run out of storage with Neon. The reason to it that in our tiered storage implementation S3 is the cheapest tier and we will offload data into S3 if it gets too big.