Thanks. The pictures really help figuring out what the issue is. In my unprofessional opinion, there could be two possibilities:
One is that this is just body oil from the previous owner staining the material. In that case, I suggest using a mild cleaner with a soft bristle brush to agitate the area to see if the oil and dirt can be dislodged. I wouldn’t use an interior detailing product that cleans and protects because you want the product to just clean the area and not leave some sort of protectant behind. This might act to just seal in the stain. In the past I’ve used a solution of Woolite and water maybe 50:50 for general cleaning of non-fabric areas. Once the area is cleaned you can use protectant.
The other possibility is that the actual material has worn down smooth from the other owner’s elbow rubbing up against it ver the years. The “grain” of the material has texture to it and it might have worn down to the point where it’s smoother and therefore shinier than the other areas. This is similar to a leather or vinyl wrapped steering wheel that becomes shiny in the spots where you mostly rest your hands. Or the keys on a computer keyboard that get the most use. Unfortunately, I don’t know of any way to restore the texture to eliminate the shininess.