Perspective is determined by one thing and one thing only: Subject distance. Period.

If you took an image using a rectilinear wide angle lens such as 17mm, which yields a diagonal angle of view of 104° on a full frame/35mm camera and cropped the resulting image so that only the center 3.08333° is in view, you would have the exact same perspective as if you had taken the photo with an 800mm lens that gives a diagonal angle of view of 3.08333°.

Why don't we all just use wide angle lenses for every shot we take and then crop the image to "zoom in" to the amount of the image we want to use? Mainly pixel density (or the size of the grains in a film emulsion). If we used the very high resolution 36MP D800 in the example above, by the time we cropped all but the center of the image taken using a 17mm lens to give an 800mm angle of view we would only be using the center 0.01625MP. That is, we would be using the center 156x104 pixels! We would only be using 1/2214 of the surface area of the sensor!

I think where the area of confusion lies is in understanding how an image projected by a rectilinear lens differs from an image projected by a fisheye lens, even if they are both the same focal length. The fisheye lens doesn't attempt to correct for the geometric distortion that rectilinear lenses attempt (and in most cases largely succeed) to correct.