At least in some systems, assuming you have a separate modem device and router, only the modem is involved in the authentication/billing which is where they care about the modem's MAC to identify you. If your router's MAC changes, DHCP would issue it a new IP.
My modest understanding is that the modem is operating at a lower layer of the networking stack, so the DHCP server involved in actually issuing you an IP, higher up in the stack, doesn't need to worry about identifying you -- it's happy to give anybody an IP because it knows that if it can see you, the modem-level authentication has confirmed you're someone who is paying your Internet bill.
I used to run an open source router a long time ago, which made changing your router WAN MAC something you could easily script.