The Router will be recognised by the Universities servers as your MAC address and then your Router will assign addresses to your computers.
As far as the University is concerned, your Router is your computer. It will also give you the added advantage of a strong Firewall, as most Routers now include one.
There is an alternative. You can hack the XP registry and fake the MAC address (make it the same as your Linux box), but it's a messy way of dealing with the problem and of course, you would only be able to use one machine at a time.