Distortion is inherent in the stitching process, whether you start with regular lens or a fisheye lens. Both have to be distorted to be able to stitch them at all.

The act of stitching flat images together produces the same overall effect as fisheye lens distortion. In order to get the images to stitch together, you have to bend them so that their edges meet. Most stitching programs I've seen will use the exif data combined with image analysis to figure out what the lens type was on the photos, taking into account the existing lens distortion as part of its calculations.

Another way to look at it: A set of stitched photos is essentially the same thing as a single shot with a fisheye lens. Think of the multiple-shot group as a "virtual" fisheye, with the capability of going farther past the bounds of what you could do with a real fisheye lens, and into the realm of a full cylinder or a full sphere.

To top that off, in order to get any curved/fisheye/panorama picture to display as a 2d flat image, you have to bend it yet again. Once you've got the stitched edges, there are various choices you can make about how you bend the image to be displayed as a flat picture. All of them involve distorting the image one way or another. The picture of the fall forest is interesting because it places the center point in the sky instead of the horizon. That same image could just as easily have been stitched and displayed as a flat/wide panorama, with the sky at the top.