You need a switch that either has 802.1x as was mentioned or the one that lets you assign certain MAC to port.
Also you can use VPN to minimize the consequences of spoofing, both users will loose connection though illegitimate user won't be able to access the resources, trafic won't be authenticated automatically.
It's not possible. They may both be using the same MAC address (by way of MAC spoofing). But they're not going to have simultaneous internet connectivity. First off, there would be an ip address conflict. Second, the switch wouldn't forward traffic properly as the MAC address would be registered on multiple ports. Third, even if the switch did work properly, traffic flow would be sketchy at best as both computers would answer for ARP requests for the MAC address in question.