Thanks Muneeb! With each reply my understanding of the ecosystem gets better.
Use case: As an end-user I want to register haseeb.id once on Blockstack, and have it available for use on Ethereum, Bitcoin, and any other decentralized network (eventually) powered by Blockstack Virtual Chains. Since it has been registered by me once, no other user on a different network should be able to register it as this is my decentralized digital identity.
If the Blockchain (and Virtual Chain) is where the domain name/public key binding is taking place, and if multiple Blockchains (and Virtual Chains and their corresponding Atlas networks) can exist, then do I have to register haseeb.id at each Blockchain independently to use my Blockstack Identity within each network?
I guess I misunderstood the role of Atlas thinking that it (or the zone files), or potentially another Blockstack layer could reject the registration & propagation of a duplicate username across two Blockstack-enabled networks.
Or, with each additional Virtual Chain supported by Blockstack, all existing domain names are automatically registered on that new Blockchain so that there can't be duplicates?