In this case, the "state" that we're talking about is "what state are you in". In other words, the router manages high-level state ("I am looking at a post now"), and provides long-lived controllers with data reflecting that state. So even though there is information about the current state in the controller (and the view), the source of truth about the current high-level state is in the router.
I've found a more specific term than 'high-level' state to explain the router state is 'addressable state'. A single page web application has two types of state. Addressable state (which is stored in the Ember router) and then another form of state, let's call it non-addressable state which is stored in the controllers.
As an example you could have a web application that has multiple documents and inside each document there are panels that can be expanded and collapsed. As you switch between documents each document maintains its own version of which panels are expanded or collapsed.
In the Ember world the router would maintain which document was opened (addressable state) and the DocumentController would maintain which panels are opened (non-addressable state).
"Application state", rather than "stateful vs stateless" state. The router specifies "I'm looking at a post now" and the controller specifies "Looking at a post means loading this particular Post model as well as these comment models"