The router recognizes a device based on its MAC and assigns an IP address. Traditionally, the MAC stays the same, so you’re right. In this case, OP doesn’t want to be recognized by the (airport) router. There is software for spoofing the MAC address for most platforms. Changing the MAC address has recently become more popular due to privacy concerns and on some operating systems it’s supported out of the box.
They glorify soldiers way too much and teachers way too little