One thing that cameras can do that we can't do with our (unaided) eyes is change focal length. When a camera zooms in or zooms out, there is a change in the geometry of perspective. When zooming in, the change is difficult to see but is well documented. Portrait photographers are well aware of this because it tends to make people look better. When zooming out, especially when zooming out to a very wide field of view, the effect can be very noticeable. When people use a phone camera to take a picture of a group of people, the people at the edges of the photo can look very flattened and unnatural. Again, this effect is well known and very much discussed and written about in photography and art. In the article about Piranesi's work, we see a very interesting remedy to the perceived distortion of a wide view. Again, these works show a field of view, and therefore a focal length, that is not possible with our eyes. We can visualise in our mind's eye what the scene looks like but what our eyes are doing is moving around the scene while our visual cortex and ultimately our mind's eye construct a mental model. In our minds, we know that the arches of the bridge are the same size and shape despite being distorted by distance. We can look at what's in front of us and see that in the receding row of arches the nearest appears to be bigger than the farthest but we also know that's just an artefact of our point of view. Things are subtly different when we look with our eyes at a photograph, painting or drawing. It's not the same as when we see something in real life, we're looking at a 2 dimensional representation of a 3 dimensional scene. We do see the perspective in the picture - we don't see a bunch of lines on a two dimensional plane. Our brains are making that happen and are making some adjustments along the way. It just so happens that something we have difficulty adjusting so it "looks right" is a very wide angle scene. That's because we don't ever see a very wide angle scene in one glance in real life.
> When a camera zooms in or zooms out, there is a change in the geometry of perspective.
This is the opposite of what happens. The perspective is exactly the same, only cropped to a narrower field of view.
Changed perspective would be counteracting the lens zoom with foot zoom -- now this causes the change in perspective you talk about. But it's caused entirely by moving the camera. The zoom is just to preserve framing and could just as well be accomplished by cropping in post, assuming sufficient resolution.
I think foot zoom is the primary reason for faces looking weird (big noses) in smartphone selfies, while "a photograph occupying less of your FOV than the camera was capturing" is why people at the edge of a photo look distorted.