When you use the Display Names addon, and someone unsubscribes from your mailing, they will see the lists that they are a member of, that you have enabled Display Names for.
When you have Display Names activated, firstly Email Marketing unsubscribes the contact from all lists -- it is just the polite thing to do! Then, its up to your contact to jump back in to an active list, if they wish.
And if a contact only belongs to one list, that contact will just see a normal unsubscribe page. Since there's no option for another list, anyway.
Here’s an example of Display Names in action. Let's say I just unsubscribed:
When I tick the box next to the list name, and then click Accept, I'll be added back to that list.
Read all about the Display Names feature.
How to set up Display Names
Unsubscribe Requests Are Handled Automatically
Subscription Preferences Addon.