However, when you're 5 miles away from the surface of a black hole, you would be inside the sun, so the shape will be different. We are not comparing equal distances from the center of mass, but equal distances from the surface; which are different distances! Only if you are the same distance from the sun as from a black hole, what you say is correct. But then you will be much farther away from the black hole than in this example.
Distance from the body I have defined as distance from its center.
Black hole does not have a surface. At least not that we know of. Event horizon is a surface, but it is not the surface of the BH in conventional sense. If you were falling into BH nothing special would happen as you crossed event horizon.
And neither does the Sun. Just like Earth's atmosphere, we can only talk about some arbitrary altitudes or pressures, but reality is that the atmosphere of the Sun or Earth does not have any firm beginning or end. And Sun is all atmosphere.
