A spiral is a symbol of the universe in many cultures. This goes back millennia.

Transformation and Rebirth: Because snakes periodically shed their skin, they are widely viewed as symbols of renewal, regeneration, personal growth, and immortality.

Wisdom and Knowledge: In many eastern and indigenous traditions, snakes are revered as wise guardians, secret keepers, or spiritual guides.

The reason a snake spiraled around a staff is the symbol for medicine is because of the healing properties of a snake.