I think when an American is talking to another American or someone they think might have a slight grasp on the geography of the American lower 48, when they say "I don't have an accent" it means they speak more or less, like they are from the west coast.

Georgia (or any southern / eastern) state is a good real life example.

I had a friend at my last base from Atlanta, and from his "accent" you couldn't tell he was from the south. Because in the city they speak with "west coast" accents. While you drive 20 miles out of the city and you can hear strong Georgian accents.

The same thing with 2 people I know here from North Carolina.

One guy is from Charlotte. You could never tell what part of the country he is from. While the other guy from the backwoods taks laik hes reel frum north kakalaka.

So when an American says they don't have an accent, they probably mean their accent is indistinguishable from the accents of those from most big cities and the west coast.