Ein/Hayate first debuted as Ein, an amnesiac karate fighter, in DOA2. At the end of that game he regained his memory as the ninja Hayate. So in DOA3 he appeared as Hayate with a new fighting style (ninjutsu) and karate was passed on to Hitomi. "Ein" now appears as a separate select option with the old karate moveset.
It is as simple as when you pick lisa and pick a wrestler costume, she appears as la mariposa, and other costumes as Lisa, same with Bass/Mr. Strong. Is just a cool nod to their wrestling characters inside the story.
Well, I knew the whole backstory to Ein/Hayate, I just didn't realize that they were two diffrent characters of him, namely,be because I really don't like him. I like the other 4 ninjas, but not Hayate. I normally jusy pay attention to the characters that I play with, or to the characters that I want to experiment with .