You are using the shield to have access to Internet, this shield communicates to the Edison through SPI, this is the reason of why you are not able to see an IP address after ifconfig -a. The chip Wiznet W5100 (from the shield) is the one managing the connection and not the Edison itself. If you would be using WiFi or Ethernet over USB you should be able to see the IP address after running ifconfig.
Thanks for the explanation Charlie.
So if I am understanding this correctly, I would not be able to use this shield to communicate with the Edison remotely from the Internet/LAN?
I was hoping to use Meshcentral or Axeda to remote manage the device, but that requires an agent running with a configured connection. So I would still need to use a USB dongle?
You can, but you will need to use the Arduino IDE for doing this. If you don't want to use the Arduino IDE, you will have to write your own library to use the shield from the Linux Terminal.
Another option could be Ethernet over USB as I have suggested above: IoT - Connecting to your Intel® Edison board using Ethernet over USB | Intel® Developer Zone