I was speaking with tech support for my cable internet provider troubleshooting a faulty router. They knew, somehow, if and when my modem was communicating with my router. Just wondering how is this possible? Can they detect the NAT somehow (how?)? Thanks.
It's possible that your modem reports the MAC address back to your ISP. If that MAC address points to, say, cisco, or broadcom, then it's obvious you're using some form of router, as neither manufacturer commonly makes NICs.
EDIT: They could also be detecting the operating system installed on the router using any form of operating system fingerprinting.