Nursing students wear long white coats too. I have seen some that do not have any name tags and patients thinking they are docs.
My hospital issued them to us with the requirement that we wear them as cover-ups outside the OR. (i.e. if you go to the cafeteria for lunch, you are supposed to have your white coat on to keep your scrubs from becoming contaminated.) Very few people wear them.
When I was a nursing student, I was required to purchase one. I don't think I ever wore it once, but it was a mandatory part of our uniform.
I get it that short white coats (in some places) indicate medical students while longer ones indicate doctors, but white coats didn't originate with physicians. I think that the problem of role confusion in clinical settings goes way deeper than uniforms, and focusing on such a superficiality does a disservice to the more important issues.
Yes, patients need to be able to identify which healthcare provider has what degree of training, and what that means for their care.
@ChrisGriffen 's thoughts about cutting off another persons clothing because he doesn't think that they have earned the right to wear it doesn't seem like a very mature response to the problem. Especially if they may not have been given much choice in whether to wear it.
EDIT: Oh, right! I forgot about phlebotomists wearing lab coats! Yeah, they were always required to wear those when they came to draw the patient's labs... you know, before they took the samples down to the.. lab.