I thought at first that the reason Diana’s relatives used snakes as familiars and had snakes on their wands was because they were deceptive, which is true, but we also see young Diana practicing magic with one of those snake wands. They seem to be a Cavendish tradition.
It makes sense when you realize that the Cavendishes are a family of healers.
We use snakes wrapped around a pole as a symbol of medical care, but it originates from a biblical story. Many of the Israelites were dying of snake bites in the wilderness, according to the Bible. When they came to Moses, he worked out a deal with God wherein he would create a bronze snake pole and anyone who looked at the bronze snake would not die of poison.
(edit: apparently it’s a reference to the staff of Asclepius of Greek Mythology; I was taught that it was a reference to the acts of Moses and never had any reason to question it.)