How are two PCs ditinguished by the game if they are behind the same router?

[EmotionEngine]

It's been a while since I did anything network and communications related especially in regard to video games, so if I post something wrong someone else is welcome to correct me.
Not only do computers have an IP address they also have a MAC address.
What's the difference? The MAC address is unique to every network device and cannot be changed (more or less impossible for 2 devices/computers to have the same MAC address)
The IP address is also unique but can and will automatically change when it's detected 2 or more devices are using the same IP address.
The modem/router also has both because it (just like your computer) is a network connected device that has to communicate with the Discovery server)
How (I think) the server distinguishes between 2 computers using the same modem/router: Think of it this way - The router is merely a post-man, he gets a bunch of letters from your router, some with your name on them and some with your brothers (Name=IP and MAC) then delivers them to the Server who can tell which person sent which letter because of the individual IP and MAC on each letter regardless of the Post-man's IP and MAC.

It's a silly example but I hope it helped.
Again if I'm wrong or not to specific someone please correct me.