He reminds me of the Japanese version of the Green Knight of Arthurian legend. His major role in Arthurian literature includes being a judge and tester of knights, and as such the other characters see him as friendly but terrifying and somewhat mysterious. 2/3 ain't bad.

I could be entirely wrong, but he does have the antler theme (not every depiction of the green knight involves him having stag horns but I personally associate him with them). He only seems to observe jack during things he feels guilty on, not acting in any way but observing, judging. That and the shows connection to various cultures mythologies.

I also do not think this green samurai is "real" more as he is a representation of Jack judging himself or his feelings of failure manifesting as an unfightable opponent

I don't know this is just wild speculation on my part, so who the heck really knows.