Dancing doesn't equal attraction. Yeah, it's the movies. Also, Harry's nothing like his father. If that's your takeaway, there's certainly some missing context in the movies. James is the embodiment of the star athlete king of the school jerk.
I was quoting Snape who said Harry was just like his father. Anyway, I agree he wasn't just like his father, but there are definite similarities in the bravery, popularity, and athletic skills. Obviously his father had a cruel streak that Harry does not have, but again Hermoine was much like Harry's mother and if you know your Freud then . . . I'm half joking, but it does happen quite a bit that men are attracted to women that are like their mother and given that Harry had no mother figure growing up, it would make sense for him to be looking for a "mother".
As for the dancing, no dancing doesn't have to mean attraction, but I don't recall seeing many women in platonic relationships dancing like they were. I know I wouldn't want my wife dancing with a friend like that. But, that could just be the movie.