The way ATproto works is all your data is stored in a database called a PDS, every post, reply etc. unlike Mastodon or Lemmy, this PDS isn't a part of an “instance” like Lemmy.world, and it doesn’t actually matter who hosts it. You could host the PDS yourself.

BlueSky is not an instance, it’s an “AppView” which is just a predefined set of rules/API, and it expects certain data in your PDS to look a certain way, and then to simplify a bit it just pulls the data out of your PDS and everyone else’s PDS all over the internet. The thing OP linked to is a Client that connects to the BlueSky AppView. But all the data still just lives in whatever PDS you store your data in. So if BlueSky disappeared tomorrow, you could write another AppView that just has to fit the same schema that the data in your PDS uses and it’d be like it never left.

To simplify further, if Mastodon and Lemmy work like Email, ATproto words like RSS. Your PDS is like your own hosted blog. Bluesky is like an RSS reader that just reads everyone’s PDS data. If the RSS reader vanishes, your data is still there, you just have to make a new RSS reader to read the data

On the AT Proto there a different kinds of data, identified by what's called a lexicon. Bluesky's aggregates data with a lexicon matching app.bsky.* (like app.bsky.post, app.bsky.like…). If Bluesky goes down you can still interact with your data, not just through the main Bluesky AppView. There are other apps that have their own lexicon (Sifa ID, a LinkedIn like, Tangled, a code forge, Stream.place, a Twitch like, Leaflet for blogging…). And even for Bluesky lexicon, there are other AppViews that aggregates the Bluesky lexicon (I think for now only Blacksky have one, Northsky and Eurosky are planning one but that's not done yet).