Many true fisheye lenses (as distinct from cheap fisheye-flavour attachments) are designed to produce a different 'projection' than wide angle lenses (that produce 'rectilinear' projection). They are quite carefully designed, and are often very complex designs. The most common type is one in which the distance the image of a point is from the centre of the frame is directly proportional to the angle the subject point makes to the lens axis. Hence the term 'equiangular'. Not all fisheye lenses maintain true equiangular projection right to the edge of the frame.
One of the properties of this type of lens is that the hypothetical 'crop factor' applies to both the equivalent focal length and the angle of view - ie if you have a crop factor of 1.5 you can divide the angle of view at full frame by 1.5 to find the angle of view with the smaller sensor.
The usual 180 degree design leads to a focal length of around 16 mm for full frame, and 8 mm for circular within full frame. The Nikon 6 mm has a field of view of 220 degrees.
These wide angles of view are impossible with rectilinear lenses. Rectlinear lenses suffer from light fall-off as the angle increases - the so-called 'cos 4th' law, though the lens design can be changed so the light fall-off approximates to cos^3rd. This fall off does not occur with fisheye lenses.
You can correct for rectilinear fall-off and for fisheye distortion in post. In some ways it is easier to correct for distortion.
Best,
Helen