If I were you, I would bring it up pseudo-jokingly and see how she reacts. “Hey, that’s weird, it’s kind of like you’re trying to be my twin! Same internship, same shoes… What’s up with that?”
Other than that, though, I don’t think you really need to concern yourself with this, because it doesn’t seem to actually be impacting your life beyond just annoying you. If she has so little identity that she feels she needs to copy you, whatever. That’s her problem, not yours. Just keep living your life.