I think what you call "same distance" is double the distance in terms of photography unless you get close to the mirror in the second shot. You are not focusing on; youirror, you are focusing on the same watch through the mirror.
This is the answer here. Most lenses distort to some level, Some more than most, others (Very few) almost nothing. Wide-angle lenses (Like a phone) tend to distort more. Mid-distance lenses like 35mm still distort a little but give a more natural look. At 44, to 50 mm lenses offer the most natural undistorted look, Closest to the natural eye. 35 is the most popular range since it is close to natural but offers a wider context. 50 is next as it offers a great natural view for contextual portraits...75 to 100 is best after that for undistorted portraits etc etc.
Longer range and Zoom lenses tend to compress the image (opposite to wider angle). Meaning you often perceive the background and subject as closer together than it actually is.
The closer the object is to the lens the more these properties are displayed. So, for example, on a wide-angle lens, the objects closest to the lens will be more distorted. This can be either pinched or pulled (distortion from the center to the image's border or distortion from the border to the center. ) Or think fish eye or inverted fish eye...
When you take the image directly on your wrist, the distance from the dial to your arm is huge in terms of wide-angle lens closeup unless you use a specialized lens. When you take an image in a mirror, you are doubling the distance to the subject, hence reducing the distortion.
In non-photographic terms, the issue is a matter of geometry and how close subjects are to the lenses (no matter what lens). This is scientifically a more accurate explanation since the wide angle lenses are closer to the subject than the zoom lenses to bring the same ratio.
See here for a comparison. Just imagine the nose is your watch and the face your arm. On the mirror, you see a jump from x to 2x
Image above is not mine. credit to Bakerdh