In principle, you could provide your own domain, and that handle would be yours to port around as long as you continue to pay for it, like any other TLD. In practice, though, most social media users aren't going to pay for their own domain. They'll use a subdomain provided by a service. The protocol assumes that service will be independent of the PDS, but AFAIK that's not required. So maybe a PDS going offline actually COULD take a bunch of accounts with it. We'll see how it's implemented.
Bluesky currently has the only active PDS, and is acting as registry for all its accounts. In theory, they'll open that up to other registry providers. For anyone who already runs a website, the simplest thing would be to bundle DIDs with their current service. And maybe that's how it'll shake out—although, it's worth noting that Bluesky uses their own DID schema, rather than the one developed by the W3C. So maybe they'll end up being the central registry for all accounts on the network!