One specific point to mention is that, for telephony, the "identity" (the phone number) gets associated with the port on the linecard in the exchange.
When you move house, and want to keep the number, then it needs to be disassociated from the old port immediately, and associated with the port/linecard where the new pair enters the exchange.
Even if you don't want to keep the number, BT is pretty keen to prevent fraud (or other loss of money) from calls made when there is no account holder. To that extent, they are keen to administratively stop the "paid" service as much as possible, while leaving sufficient service to test the line.
Broadband is a little different. Each line terminates in ports in different hardware in the exchange building. And, so long as the old/new ISPs use a BT wholesale service, then nobody cares *which* port the line comes in.
With broadband, the "identity" is discovered after the line synchronises, when the ID and password are used in a PPP connection. Packets are then switched after the PPP session is authenticated.
That difference in identification and authentication is the significant bit.