If you have a sufficiently wide field of view, even the standard projection transformation gives you things that look pretty "distorted", which is a fun trick often used in art photography. Also fisheye lenses are often designed to do a non-projective transformation and get effects you can't theoretically get with a pinhole camera, like mapping straight lines to curved lines as Pomax does.
Pomax's invention is super cool, even though it isn't the usual 3D→2D mapping motivated by pinhole-camera geometry. Weird is good! I wonder if there's a version of it in which straight lines become curves, but polynomial curves of some reasonable degree, or maybe something involving square roots and division, so we can rasterize them without computing a bunch of exponentials? Is that even a reasonable way to seek optimizations in 02021?
Pomax may have a good point that projective transformations aren't "strict three-[vanishing]-point perspective" because, instead of having three vanishing points, they have an infinite number of vanishing points, a point others have made in the comments here. The profoundly weird and cool aspect of Pomax's invention is that every line converges to one of those three points.