As a title, Tevye is pretty standard as far as the anvil-like books of realistic literature go: Jane Eyre, Oliver Twist, Tess of the D'Urbervilles, Anna Karenina. You get the point. Sholem Aleichem is not rocking the realist author boat here.
But check it out. The title isn't actually just Tevye. It's Tevye the Dairyman. Now, "dairyman" is only one of Tevye's roles. Why not Tevye the Father, or Tevye, Golde's Husband? Or what about Tevye the Learned, say, or Super-Unlucky-God-Sure-Has-It-in-for-You Tevye?
Here's one theory, and we think it's a pretty good one: the main difference between just saying Tevye's name and saying his name and his line of work is that the title puts him in the economic structure of his village. He's not just some dude named Tevye. He is an integral part of the way that little microcosm functions. You want to drink some milk? He's the guy you want. Fancy a grilled cheese sandwich? Go see Tevye.
The upshot of the stories is that regardless of how deeply rooted and embedded Tevye is in his community, his neighbors have no problem at all going along with the forced resettlements ordered by the government. So, it's key to right off the bat point out just how much a part of normal everyday life Tevye's stories are at first—so it's an extra kicker when it turns out that they're really about the destruction of Jewish life.