Boy, this is a topic.
The ratio of image sizes of two or more objects in a frame is directly proportional to the object distances. This however does not describe perspective, or only in comparing image from the same format.
Perspective, the apparent depth in the 2-D image, is a little more difficult. Given the same field of view and object distance, the perspective will be the same. This is the idea of equivalent focal length.
However, cropping or changing the focal length while maintain the object distance will change perspective. This is because the image to reach its final display size will be enlarged proportionally more. You can look at to two ways, either the foreground and background are not the same in the uncropped and cropped image meaning the ratio of object distances to the foreground and background are different in each image and so have different perspective, or the infinity points (used in linear perspective) are displaced giving different perspective in each image. So a 50mm lens on a 35mm camera will create an uncropped image with a different perspective as an uncropped image made with 50mm lens on an APS camera. Don't believe me? On the left is an uncropped image, to the right is a crop taken from the image and enlarged to the same size. The red line clearly show perspective has changed. (Yes, if I reduced the crop in size it will match the set on in the uncropped image, but we do not scale images to some absolute scale based on FoV and format size--magnification is the bit folks forget.)
BTW, while I am sure folks will be shocked, but this is not new nor radical. This is how it has always worked, but folks have been confusing the ratio of image size to the ratio of object distance as a rule about perspective for a very long time when there is not a direct link.