Two weeks ago I went to see my Doc. When I took off my shirt he noticed the empty holster on my belt (the clinic of course is a gun free zone) he asked me what I carried, I told him and he said that his carry choice was a Glock 19. After he completed his examination we had a short conversation about the relative merits of conventional DA/SA pistols vs striker fire. I somehow doubt that he will be passing any information on to "big brother".
Well, your doctor may not really have much of a choice. Part of the requirements for certified Electronic Medical Records includes automatic sharing of information with "assigned" government agency. Not sure if the actual implementation has started but the design is clear and if you make EMR software you are required to demonstrate that such a real time feed is functional.
Also, with many of the doctors now closely associated with a hospital chain, the hospital chain actually stores the information and configures that actual questions required. So, the nurse that you see either supplies an answer or you supply an answer. Now, how that information is used is anyone's guess.
I think it is a good thing that collection of non-medical information by doctors is limited by law.