That is not a "Router" that is a "modem". Two different things.
Modems are only made to connect one PC at a time to the internet, and they tend to lock their settings in to a particular network card using the MAC address. If you want to setup your new computer on the internet, you need to do a reset of the modem to clear it's saved settings, and then hook up the computer to it, and reboot.
If you want both computers on the internet at the same time, you need to get a router and hook that up to the modem, the computers get connected to the router.
the device he has is a Cable-Modem/Router one of those all-in-one devices