There is a complex system of multiple authentication steps in the ADSL connection. The first login (configured in your router) is just the first step. There are several more along the line, all within the BT ATM network, on the way to the ISP's premises.
You can get funny things happening if you have a dispute with your ISP (say, he cuts off your account for alleged P2P abuse). Under the BT wholesale terms, BT will prevent you signing up with a different ISP until your dispute with the previous one is resolved. Crooked ISPs (I know a few) will use this to bully customers into paying for a service which was not provided, etc. And BT won't help.
Unless you ask BT to change the phone number of the phone line! That kicks off the previous ISP and they can't prevent it. But there is still a delay of a few days while BT sort out their stuff; the old ISP still "owns" a part of the connection within BT.
Really weird.
Any business that relies on an internet connection would be well advised to have two analog lines, so if there is a problem with the ISP they can get another ISP started on the 2nd line concurrently. Otherwise, the old ISP has got you over a barrel. Nowadays many businesses are on ISDN which is no good...