A fisheye’s projection is not caused by one special “fisheye-shaped” surface alone, but by the overall optical design.

With a pinhole, projection is purely geometric and rectilinear. Real lenses are different: once you use thick, multi-element optics, the image mapping depends on many interacting variables, including element curvature, spacing, glass refractive index, and element shape. Those choices affect aberrations, distortion, and projection.

So a fisheye is designed to intentionally use a different projection mapping than a rectilinear lens. A rectilinear lens is actually something designers have to work to achieve; it does not happen automatically just because lenses use ordinary spherical surfaces. Aspherical elements are also not what makes a lens inherently “abnormal” or fisheye.

In practical designs, fisheyes often have a notably different front-element geometry from comparable ultra-wide rectilinear lenses. For example, the front group may be shaped so the lens is much thicker near the edges than the center, helping produce the fisheye image mapping. But that shape works as part of the full lens formula, not by itself.

In short: fisheye vs. rectilinear projection comes from the complete lens design and the projection the designer chooses to optimize, not from a single simple lens profile rule.