Why does CBS All Access only work on my computer but not on my TV?

I subscribed to CBS All Access through Apple TV on my laptop. Afterward, I used Apple TV on my TV to access the same content and it acted as though I was a totally new user. Same as when I tried to log into CBS All Access through the CBS app. So how does this work? I'm paying a monthly fee to only access CBS on my laptop and only through the Apple TV app? That seems awfully limited for a service promising 'All Access..."