Nope.

Your handle consists of your alias and your home instance. ie, "@sohcahtoa82@nerds.social"

If I want to move to another instance, say, jocks.social, then my handle would become @sohcahtoa82@jocks.social, and that's assuming @sohcahtoa82 wasn't already taken on jocks.social.

So it doesn’t help much to just move your data if you get banned by a mod, does it?

Somewhat correct. The challenge becomes informing your followers of your new account.

Instance mods can block entire instances, making it possible for an instance (or set of instances) to become isolated from the rest of the Fediverse. At that point, those instances might as well be a completely different site, since they'll no longer communicate.

And I somewhat expect that to happen. Many instance admins will block instances that become known to be cess pools of hate speech.

Your old one doesn't go away unless you delete that account. You don't automatically get to keep your old handle on the new server because it may already be taken (handles aren't unique across the entire network), but if it isn't you can keep it by normal registration.

It's like email. You can keep your username if no one else is already using it on the new instance/mail server. The part after the username is different for each instance/mail server, like hotmail.com and gmail.com email addresses are different instances.