Technically you only need to register in one instance and can interact across lemmy and kbin instances, but you can register the same username in various instances in case of defederation. Lemm.ee federates with all and all instances federate with it for example, beehaw.org doesn’t allow signups and has defederated lemmy.world and sh.itjust.works.

That aspect is like email. If I have the email address Limeade @ protonmail.com, someone else can sign up for Limeade @ gmail.com without it affecting me. Gmail doesn’t ask all the other email servers around the world if anyone is already using Limeade. If you wanted to send me a message, you can’t just send it to Limeade @ anything.com and expect it to reach me.

Depending on which service you are using, sometimes usernames show the full username @ domain.tld format by default, others just show “username” and you have to click to see the full info that here on Lemmy and the wider threadiverse I am not just Limeade, I am @Limeade@beehaw.org (the extra @ in front is the standard format for writing usernames on the fediverse).