It is a TV show from 1999 that centered on the literal office of the same name that the US President holds court in.
The West Wing is not the President's office, but rather the section that contains The Oval Office, which is President's office, along with other executive offices and rooms, such as the Press Briefing room, the Situation Room, etc.
And I have no idea what "The West Wing" is. And I doubt I am rare here at all also.