Unless the object is flat, say a coin or a medallion, and the background is empty, say a flat colored background, the photos will not be the same.
It's just like portrait photography, with the nose looking too big relative to the ears when shooting from a short distance - perspective works for the front & back parts of the subject same as it works for the subject itself being in the foreground and other stuff being farther away in the background.
Therefore, if you shoot a diamond ring with the diamond facing the camera and the band away from it, the shorter the focal length the larger the diamond will look compared to the band.
This is actually the answer that I was looking for when I wrote the question. I had a hunch, but probably could not put my question properly. Thank you.