The camera, if standing at the same place, can only see the same one view of the subject. We have to stand some other place to see a different view. But from wherever the camera is standing, that is what the view looks like. Meaning in a portrait, if the nose too near appears enlarged too much, it will look the same at any focal length, from that place.
If standing at the same place, the relative size of a distant background object will appear the same relative size at any focal length (so I am also ruling out compression, it is just perspective). However, the subject size, and field of view, and the depth of field, certainly do vary with focal length, if still standing at that same place (these are magnification properties, not about perspective). But the view of any one smaller area does not change when simply enlarged. However, when you change where the camera stands, then certainly the view changes too.
We do see comparisons online showing portraits at different views of short and long lenses, but of course, they necessarily stand in very different places to make the subject be the same size in all. That is a perspective difference, but it is 100% caused by the view at that distance, not by the focal length. Focal length does cause us to move where the camera stands, but the view we see there depends on where we stand. Focal length is just a magnification of whatever view is seen (we like magnification too
)
It is a fairly obscure point. In practice, we necessarily do have to change where we stand with different lenses. The choice of the right lens does allow us to stand where we think is proper - or allows us to get a proper field of view from where we must stand. But if we don't move, then the view that we photograph is the same from that one place.
If you don't move where you stand, and if you crop the widest field of view (short lens) smaller to be the same as the narrow field of view (long lens), that corresponding area will see the same perspective. The really big difference is the cropped view has far few pixels remaining, the cropped view is a much smaller image size, requiring greater enlargement, not capable of the largest prints the way an uncropped image could have been. Using the longer lens crops the field of view too, but it still uses the full sensor size. But cropping it later discards many pixels, and we only have a smaller image remaining, so resolution is seriously reduced. If you already had an excess of pixels, more than you need, this may not be a big deal. If you didn't, it often is.
We always want to stand back 6 or 8 or 10 feet for a portrait (for proper perspective). This is true of a group shot with wide angle or a head shot with a telephoto, standing back a bit is important for proper perspective. But then if you crop out only one head in the group shot, its view will look the same as the longer head shot (if still the view from the same distance), except the cropped image has become a very small image, not capable of becoming the same good enlargement into the same large picture frame.
Saying, if the wider lenses has to stand too close to achieve size, that ruins the perspective of a portrait. Standing back and cropping can help that, but then again, it becomes a smaller image with less resolution. It is why we have different lenses, or zoom lenses.