Jax is obviously the ninja turtle Raphael, accidentally brought to Runeterra across time and space and doing his best as a soldier-for-hire to stay alive while finding a way to return home.
...and I mean the bad-ass original Raphael. The one with the giant ego who would use anything (including the Shredder's spare armor on mobsters) to win a fight. He was also well known (as were the other turtles) for ending up in other dimensions at random through all versions of the series, one of which at least saw him donning an entirely black spandex costume to wrestle in a cosmic arena.
I always thought of the symbols on his cape as turtles, and that slightly suspicious way his cape seemed to be covering more than just his shoulders and back.