You can think of ATProto like this.
Each user has a "website" with JSON of their own content (e.g. all my posts, all my likes, all my follows, actually live in a sqlite database hosted somewhere). It's not really a website but more like a git repo — one per user.
And then, there's a protocol for how to aggregate information from all such "websites" in the network into a stream of changes. Apps subscribe to that stream of changes and update their local databases (which act as app-specific caches) in response to those events.
When I make a Bluesky post, I'm really writing JSON into my sqlite file. This change gets broadcasted to all interested apps which update their own databases (which may or may not care about specific content type like "Bluesky post"). Obviously forks of Bluesky backend do index Bluesky posts (and then return them in the same UI), but you could imagine other backends that only care about other content types, or that record Bluesky posts but in a different database structure, and ofc can present a different UI for it.
Yes, you can run your own node — multiple types of nodes. You run your own PDS (https://github.com/bluesky-social/pds) to store own data (that's the "website" in my analogy), or you could run a Relay (https://whtwnd.com/bnewbold.net/3lo7a2a4qxg2l) that collects all PDS changes into a stream, or you could run an AppView (any backend that listens to Relay or PDS, basically your own app).