Arduino Ethernet shield cannot establish connection
The W5100 chip doesn't have a mac address by itself. You can assign any mac address to it that you want. With Ethernet.begin(mac); you assign the mac address to the W5100 chip.
The official Arduino Ethernet shield comes with a paper with a unique mac address. That is a nice service, but that is just a number, the Ethernet shield hardware is without any mac address.

There has been complaints about wrong resistor values near the ethernet connector. Can you read the numbers on those resistors ? They should be labeled as "510" and not as "511" : Arduino forum about ethernet resistors.

The Ethernet Shield uses more power than most other shields, and the W5100 can get pretty hot. Do you have enough power ? Is the 5V pin still above 4.5V ?

It is possible that your router does not accept the old W5100.

If the resistors are okay, I suggest to try an example with both mac and ip: Ethernet.begin(mac, ip);