Can anyone explain what the zone off the coast of The Outer Banks is?
"North Carolina W" but has no information with it.

You might try airmap.io

Looks like it's actually more closely shaped to a rectangular area - it's labelled "Special Use Airspace" W72A (North Carolina W, NC)

It's all off-shore, but unfortunately, the DJI map can only show zones in circles, so it appears to include the shore but it actually does not.  

There's also a drone flight notification at the beach showing in airmap which I've never seen before!  Is that you?  Just a tad south of Corolla.  It says

Flight
Start Time: Oct 14, 2018 10:00 AM
Flight Time: 24 minutes
Radius: 250 feet
Altitude: 350 ft
Aircraft: DJI - Phantom 4 Pro

No, just getting there now. I would guess someone has done an unlock, but I've never seen that before either. Imagine how the map would look if it showed every drone floght.

Ok so I searched for a sectional chart and found the answer. W72A is a military operations area. As The Mark pointed out it is rectangular and starts well offshore extending over the ocean.
I also talked to a Kitty Hawk police officer who told me they don't have any local laws against drone flights.