Yes with sky, there is no such thing as your personal username and password. The username and password is derived from the MAC of the router and will work on any Sky connection out of the box.
That said, there are plenty of routers sold on ebay which do not work. This is largely because somebody gets the sky router, extracts the username & password, uses their own equipment, then sells the Sky router on ebay. Of course now, there are two users attempting to connect to Sky with what appears to be the same details, as such there are issues.
Also if a user does not make a payment, sky have been known to then go into the users account, find out the MAC of the device they believe they are using, go into the list of allowed devices on their network, & put a manual block on that specific router.
Effectively, if the router is not on a block list (ie seller not made payments) & the seller is not using the same details (ie own router with the sold routers details), it will work.
In terms of who is accountable for the internet usage, it would always link to the correct user, ie the buyers account, e.g. illegal downloads etc.
EDIT: When a device sold on ebay is used by another customer, Sky will do a check that the router is not barred, and is in the allowed pool, then they will register the new MAC to the new users account. Process takes around 1 hour to complete. If it is network blocked, it will not work.
Edited by ukhardy07 (Mon 25-Jul-16 00:18:26)