Doctors aren't assigned in Canada, unless you're referring to the doctor on duty in an emergency room. A person is always free to request or seek another physician. And as to formal complaints of incompetence, I believe they're all looked into, however the College of Physicians has the unfortunate reputation of being pretty lenient on many of their members. But, sometimes they do take actions as confirmed by the resulting case of Dr. Charles Smith, former chief forensic pathologist for the Province of Ontario. Many, many lawsuits pending because of this guy's incompetence.
Of course, another doctor might not be available in the middle of the night in an emergency room. In that case assuming the emergency isn't dire, one can walk across the street to another hospital, which is something I've done after four hours of waiting. Triage is a bitch sometimes. Going to another hospital is easier to do in a place like Toronto, but relatively difficult when you live up north where another doctor or hospital might be many hours away.
And family doctors are getting increasingly hard to find in Canada. The USA keeps thieving our doctors by offering better pay and few working hours. Guess I can't blame them for that, but I think doctors graduated in Canada should be required to commit a specified period of service in Canada before they move lock stock and barrel to sell their Canadian acquired skills elsewhere.