I thought it might be an interesting idea at first, as I kind of like the names of the traditional archetypes that the Elder Scrolls games have employed in the past, but I think, in terms of realism, that just being called "Dragonborn" would suffice. As others have pointed out, you really can't tell exactly what someone's "class" is going to be just by looking at them, so it wouldn't make sense that an NPC would automatically know. I mean, most of us are going to have a sword... but that doesn't necessarily make us a straight up "warrior" type.
My mod wouldn't determine class by looking at your physical appearance or equipment. NPC's wouldn't verbally recognize you either as this would require voice acting, something I'm not going to add.
I would, however, like to have some kind of "class" recognition on my character sheet, based on your idea of the game recognizing which abilities you use most and then assigning you a non-static class name and displaying it on your statistics... WITHOUT an NPC being able to call it off. I just want to kind of know that it's there.
Now you're starting to get it
All it would do is look at your skills and, based off the percentage each takes of your overall skill-set, would determine a class. It might look at your spells and perks, too, for further depth.
Fable had this system, and it didn't exactly work.
How did Fable's work? I doubt my mod would be anything like it.