I contacted Amazon Technical support. They stated the "Network port on the back of Echo Show is meant for specialized Enterprise experiences. We currently do not offer support for consumer use of the Network port."
I asked for documentation on the "specialized Enterprise experiences" and was informed "that information is not available on Amazon website. It is internal information. We do not have access to further details regarding this port."
This is sad considering many media sources and reviews touted the port as a feature.