I did some research on this and I am going to share the wisdom I acquired, with you.
The story begins with Hippocrates, who is considered the father of medicine.
Hippocrates began his healing career in Asclepius, which was where there was a temple used for healing. At the temple, snakes were a part of the healing rituals. Also, snakes were symbolic of death and renewal due to the ability of the snake to shed their skin and create a new one, which is also representative of the healing process.
But what about the stick?
Well, it is actually a staff. It is said that a staff was carried by the Greek god Hermes, who was a messenger between the gods and humans. He was also the patron of travelers, which in the olden days before hospitals, doctors had to travel great distances by foot in order to visit their patients.
So when did it become a logo?
Well, in 1902, an officer in the Army Medical Corp insisted the snake and staff be their insignia. Now, I have been in enough meetings to know there are probably some in the meeting who don’t even know what an insignia is, and the rest don't care. So most likely everyone said, “OK, if you feel that passionate about it” and then someone said, “Meeting adjourned”.
But as more business and professions expanded, there was a need to help people easily recognize what type of business you were in, so an easy-to-understand picture was needed, especially if not everyone reads. I can imagine the boss telling one of the clerks to research what other people are doing in the medical field and the finds there are not many logos to choose from, but he does find an example of the Army Medical Corp insignia and takes it to his boss.
The boss probably said, “well, if its good enough for the Army of the United States of America, it’s good enough for me!”
So the use of this insignia with the snake and the stick grew, Now days 62% of all medical companies use the snake and the staff in some form in their company logo.
So that is how the snake and the staff became part of the medical profession logo.