black holes - Is Hawking radiation really the same as Unruh radiation

I read that Hawking radiation is the same as Unruh radiation. However, there seems to be a paradox here.

If you have an extreme black hole (say with maximum charge), then it has temperature 0 and doesn't radiate. However, it seems to me that a neutral (uncharged) observer hovering above the horizon should still see Unruh radiation because he is undergoing a high acceleration.

Does this show that Hawking radiation and Unruh radiation are really different things? If not, how does one resolve this discrepancy?

And either way, doesn't the observer hovering over the horizon see the Unruh radiation escaping from the black hole? If he sees this, why is he wrong?