Hmmm, I was thinking about this when I was reading the OP's comments. I know a few phd's at my hospital who have said the same thing to me, come by my lab/call me/stop by anytime, etc. These are people who were also my professors. A few MD's have said the same thing.
I am an older student, and so some of these people I think, would sort of like to become friends. I mean, like the phd's, and one or two MD's that I can think of. So, I also sort of agree with the OP, that when you sort of start to get to know someone personally, and yet that person is also your superior in some way - well, it gets hard to figure out the social dynamics.
I know an MD, and like him personally and socially. I think he likes me too, and is well, sort of becoming a friend I guess. And yet, he is a very senior physician in his department. I will eventually have to rotate through his division, altho I elected to do the rotation at another hospital, because I felt the dynamics were complicated. We're just friends tho, and yet - I don't know. It's hard to sort it all out.
So, I agree with the OP, it's hard to know were "student" is separated from friend stuff. Just trust your instincts and be yourself is my recomendation.