Yes an Anesthesiologist is a real doctor and is an MD or DO who mainly practices and is qualified in Anesthesia.
The main job of an Anesthesiologists is to specialize in perioperative care, developing anesthetic plans, and the administration of anesthetics.
But they are real Physicians and real Doctors and go through the same training as a regular doctor but they also go through more training and studies than a regular doctor so that they can qualify in Anesthesia.