The blue doctor symbol you're likely referring to is a blue caduceus, commonly used as a symbol for medicine and healthcare professionals.
While the caduceus (a staff with two snakes entwined and often winged) is often associated with medicine, it's important to note that its use is sometimes debated and confused with the Rod of Asclepius (a staff with a single snake). The Rod of Asclepius is the more accurate and historically appropriate symbol for medicine, attributed to Asclepius, the Greek god of healing.
However, the caduceus has become widely adopted, particularly in commercial contexts, and is often presented in blue to denote trustworthiness, professionalism, and health.
In summary, while technically the Rod of Asclepius is the more accurate symbol, the blue caduceus is a widely recognized, albeit sometimes debated, symbol representing doctors and healthcare.