As others have alluded to, if you’re from the region, one doesn’t generally describe it as “region-style” foodstuff. Just like I wouldn’t be called “Chicago Pete” here in Chicago, but I might if I were living in, say, Minneapolis to disambiguate me from the other Petes.
And that’s most likely because they’re targeting an audience beyond NYC. Just like we have “Chicago style cheesecake” here in Chicago, but we (as far as I know), don’t really refer to it as “Chicago style.” Or “Chicago style pizza,” but here it’s just deep dish or stuffed pizza (depending on which thick style you’re referring to, or just plain “pizza” if you’re talking about Chicago style thin crust), unless you’re from elsewhere.
I agree with you completely… it doesn’t mean that the other 99% of the planet doesn’t refer to it this way though. And when someone gives an example of how “even Junior’s” doesn’t describe it this way"… I just had to show them otherwise. I’m not sure if it is being a pedant, or an asshole, or a combination of both… but I just couldn’t let it go unchecked. 
Ahem, “San Francisco sourdough” begs to differ.
I chose apple pie.
But if I could add a fifth; bread pudding. Especially Golden Corral bread pudding.
Nathan Detroit: Still, you will admit that Mindy’s cheesecake is the greatest alive.
Sky Masterson: Gladly. Furthermore, I am quite partial to Mindy’s cheesecake.
NOW you’re talkin’.
Ergo, “generally.”
Indeed. No argument there.
You can top a NY cheesecake with all the other ones… 