The logins aren't federated, the content is. Each instance receives a copy of everything, and normally you browse other instance's content from your home instance. In your case you'd access lemmy.one's content from your home instance, lemmy.world. If the logins were federated we wouldn't need those domains after our usernames!

The email analogy still works for this: if you're on Gmail, you don't go log in to Outlook to send an email to your friend: from gmail directly, you send an email to your friend and Gmail's server takes care of sending it out to Outlook.

There's browser extensions to help go back to your home instance, as the linking on Lemmy is sometimes a bit weird and you do end up on other instances every now and then.

Your account will only log you into your home instance. To interact with threads on other instances, you'd need to locate the post from within your own feed on your home instance.

Aren’t lemmy logins supposed to work across the whole federation?

No, they are not. Every instance is supposed to have it's own logins. Everything is working as intended.

No, this isn't how this works. Each instance manages its own users in its own database. All that is federated is the posts and comments. You should be able to post and comment to other instances' communities but if you want an account on another instance you have to create one.