Harry was in costume. Tuvok/Chak and the EMH weren't because they were on duty looking for their missing crew, not playing.
I think the EMH was programmed differently, therefore he had what he perceived was emotion at this early stage. He was programmed with " two thousand medical reference sources and the experience of forty seven individual medical officers". I suspect that experience includes a fair amount of "emotional" bedside manners. Some good and some not so good. That bedside manner allows him to relate in a way Data never could.
As for relating like Data... a mild spoiler is the EMH isn't "fully functional".