The proxy server does a great job of isolating pcs from the internet.
It probably has rules to accept only known pcs, either by MAC address or ip number.
If this is in a University setup, I would expect there to be several other controls as well..
eg HTTP headers, TTLs & passive tracking.
There is a lot of pc specific info sent while browsing..
Connecting to the proxy server through another pc is possible, it's called masquerading
That would need to be set up on the intermediate pc..