Greenwich Village is “The Village.”
See post #33.
When somebody mentions the War, they’re usually referring specifically to World War II.
I-84 through Portland, OR is referred to as “The Banfield” on traffic reports, although it’s not used by anyone else. Looks like the name is referenced on Wikipedia.
John Wayne was “The Duke.”
George Steinbrenner was “The Boss,” except in New Jersey, where it’s Bruce Springsteen.
The Stuff The Stuff (1985) - IMDb 
Not unique by any stretch, and a bit of a curiosity: The Zone.
Almost every station in this area is “The ____” something.
I propose another thread to locate the most ridiculous name for a radio station in The World!
As is Gibraltar I believe.
The Hague.
This is actually a Southern California practice. Just take the 101 to the 405 and get off on Santa Monica Blvd.
As opposed to take 280 to 101 and go north.
How many Hagues are there? This is how I ran afoul of the OP’s premise.
There is only one “The Boss”
Similarly, though not necessarily rural, I’ve heard people refer to someone that died of ‘the cancer’.
That’s correct – items which already have “the [blank]” as its official name, or are too inspecific (i.e. “the Border”) don’t count. Otherwise this list (aka “The List”) would get way too out of control. 
Also, I’m afraid I’ll have to call foul on “the Ocean”, since the term doesn’t specify any particular ocean (and one could argue that there is only ONE ocean anyway.)
And not from fiction, Sutherland Shire in Sydney is known as “The Shire.”
Though it’s been a while, I’ve also heard rural people call it “the sugar.”
… there are other The Hagues besides The Hague? 
The Chairman of the Board - Sinatra
The Big Game - Cal vs. Stanford
No there aren’t and that’s the point to defining the scope of this thread which I caught on to belatedly.
Darn Straits.