I get that the vet super saved his ass, but how does that dog have a concept of "this guy performed life saving operations on me"?
Unless he also nursed him back to health.
He did. Him and his entire team did, for weeks on end. They managed his pain and taught him how to trust humans again. Related article: http://pitbullnews24.com/2018/04/19/dog-found-with-severe-burns-reunites-with-the-vet-who-helped-save-his-life/
more info: Smokey the dog was chained to a tree when a fire broke out at the homeless facility where he was housed. He had severe burns all over his body, was brought to an emergency vet center were Dr. Latimer (and the nurses that cared for him) were able to save him. Before long, he was well enough to be adopted by a family. The gif shows Smokey seeing Dr. Latime after several months for a check up.