I am the Kat that Walks By Itself

"Miss Granger" is what he called her when she was a student, which indicates an unequal relationship, and emphasizes the fact that he was her teacher.

You might notice that the students call each other by surnames, "Potter" or "Malfoy," for example, and while it's true that seems more male, it's also true that adult equals in situations like the military, police force, etc. often call each other by surname regardless of gender. For that matter, there were several friends of mine in high school who sometimes called me by my surname.

So it depends on how the writer wants to use it, but in general I think in ss/hg, "Miss Granger" is not treating her as an equal, "Granger" is treating her as an equal but perhaps not particularly friendly (though that can vary with context). Hermione may or may not be friendly/affectionate, or may just be matter-of-fact.