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.
Someone will be desperate enough to work there. Now McDonald’s has a competitive advantage and other places will have to follow suit and then it‘s worse everywhere. It‘s amazing that there are places out there where regular people can actually influence anything.