If an ISP sets up cable modems in bridge mode and lets users change the mac address on their internal router (or completely change routers), which then assigns a new IP address, how can the ISP track who has what IP address at any given time? Are there additional unique identifiers sent from the cable modem that allows the ISP to tie all traffic to the customer regardless of mac address or IP address?
That depends on the ISP. Some ISPs use PPPoA or PPPoE where you log in using a username and password. In those cases it is easy to identify the customer. Some ISPs provide fibre to the home and use identifiers attached to the ethernet ports to identify the customer. I'm currently working on software to manage such identifiers for IPv6. And yes, cable modems are also identifiable by the ISP.
How this identification affects address assignment again depends on the ISP. Some ISPs just assign you fixed/static addresses where your customer identification is linked to your IP addresses and other ISPs just let their DHCP servers assign some addresses to the customers and only use the customer identifier for logging and legal traceability requirements.