The modem has to be authorized via MAC address on the network. They know the MACs of the modems they own, so it's very simple to know if a modem is theirs or not.
That has nothing to do with what he said. At the very least, you can leave their modem also set as a router and just put your router behind it (double NAT). More likely, you can turn off all of the "router" functions of their modem and still put your own router behind it.
There is no way for them to differentiate whether the device behind the modem is a router or a computer.
There is no way for them to differentiate whether the device behind the modem is a router or a computer.