There are lots and lots of studies that show that, basically, people are more likely to trust a traditionally dressed physician who acts in a certain way. Kakhis+collared shirt+white coat > scrubs > anything else. No piercings > piercings. No tattoos > tattoos. Smiling > not smiling. Sitting in the room > standing in the room. So in that sense you are right: physicians do best on satisfaction surveys when they look like a 1950s stereotype of a physician.
That does not, however, mean that you always have to do every single thing that makes your patients most likely to trust and like you. You have worked hard enough to earn some discretion about how you look and act. I am also in the military, and I think the military has so many retention problems at least partially because they can't distinguish between 'this is a good idea' and 'this needs to be mandatory for everyone'. It is a good idea to not wear a nose ring. It is a good idea to scrub your hands before a C-section. However only one of those things is a sufficiently good idea that it needs to be mandatory for all residents.
As a physician, it is mandatory that your patients like you to at least a certain extent. Enough patient complaints will always get you fired, particularly as a resident. I personally try and do everything the studies say to make myself more likeable, because I consider those easy layups, and they offset the negative impression I make when I don't give out antibiotics for viruses or when I lecture parents about why vaccines shouldn't be optional. However I also don't have any particular emotional connection to the way that I look, and don't particularly want a tattoo or nose ring in the first place. If I did I would probably find another way to be likable.
Your resident knows what the nose ring looks like and apparently its worth it to her to keep it in. If she is still a resident I am guessing she comes of as professional enough that the ring doesn't tip the scales. So don't worry about it.
