If memory defines the person like some people seem to be suggesting, does that mean if I forgot what I had for breakfast I am somehow less of a person? I don't remember everyone in my 1st grade class, I've lost something of myself? That a person who can remember every trivial detail is somehow a greater person? Where's it stop?
Kurn's new persona knew how to speak, how to walk, so one assumes he has all his skills he had even if he doesn't remember them. I don't remember who taught me how to work a light switch, but I still can do it quite well after all. He was obedient and respect towards his "father." Kurn worshiped at authority's shrine his whole life. He'd follow orders without question and expected obedience from those below him. He expressed curiosity at this other Klingon who happened to be there, suggesting he hadn't lost his will and drive at all.