Those are just lies told to the residents of 101 to keep them afraid.... The Pip-boy can be removed by anybody with the right tool. The sensors are a whole nother issue, of course, but you could likely expect that they simply aren't as sensitive as they were 200 years ago. They are still capable of picking up your heart rate and the such (consequently capable of telling you your HP and other stats) but that they are no longer capable of differentiating one user from another, beyond that.
If the Doc knows how to remove and re-install a Pip Boy, it's reasonable to assume he also knows how to reset the sensors so it can be calibrated to a new user. Maintaining a Pip Boy is not a Vault Doctor's job, but thier job...and that of the Pip Boy Technicians...would be made easier if they were trained on certain basic functions that overlap thier duties. Anything beyond removing it from a patient, putting it back on the patient and running checks to make sure it is working properly would normally be referred to Maintenance....but I don't see how someone trained to install a Pip Boy and make sure it is reading the wearer's vital signs properly would not have been trained to reset it or not have access to a service manual that explains how to do it. Even if such limited cross-training was not Vault-Tec doctrine....the Pip Boy Techs would learn they could save a considerable amount of time and effort by training the Doc unofficially. It's not like he is a threat to usurp thier jobs, after all.