Instances do not care about users on other instances like that.

A user is not unique by username alone. A user is unique by username AND instance. There may exists another skye on another instance, created by someone else. But only you can be skye@lemmy.world

Think of it like email addresses.

You grab the name abc on gmail, giving you the user abc@gmail.com

Some other person can grab the same name on another mail provider (ie. instance), say abc@outlook.com

You don't need to have an account on every email provider to be able to send mail accross providers. But nothing is stopping you from making an account on every provider, it's just that it's redundant