Hi Jan,
In the first photo, the hands and head are on roughly the same plane- ie the same distance from the camera. Their relative sizes look convincing.
Look at the second picture, where the hand on our left looks extremely large compared to both the head and the hand on our right. This distortion is pretty dramatic, given that the hand on our left might be as little (or, in fact, as much) as 12" in front of the hand on our right; the distortion comes from being too close to the subject for the lens you are using.
The "portrait" setting refers to aperture, or how large the lens opening is when the shutter is triggered. Aperture controls the depth of field, so that the face can be in focus and the background fuzzy, but doesn't have an impact on the type of distortion in your photo.
Search the Forum for topics like
"focal length", or "photo distortion", to try to maximize the quality of photos for the camera you have.