Customers
You want to see the person, not only the anonymous chat. Customers is that list: name, contact details, and the conversations that belong to them.
It stays empty until your website tells Cusu who someone is. A visitor can still chat. They simply are not on this list. That is normal for a new group, and it is normal for anyone who never signs in.
When your site does identify them, later conversations attach to the same person, so you are not guessing whether this chat is the order from last week. How the site sends that identity is a developer task. It is described in Identify.