Well ... the dark paneling nosing is so old, why not rip it off and update your decor? You can easily do a quick stain match (Minwax) and rub it into the scrape. The stain comes with sealer included.
The door is worse ... color and appearance and I can't see the depth of the scratch. If you want to be fastidious ... pull the door and strip the whole side twice with Jasco paint stripper. Sand the entire door, especially the scuff, to smooth. Rub down the entire door with 000 steel wool soaked in paint thinner, wipe hard with a soft cloth, and let dry for at least 36 hours. Rub down with soft clean cloth to remove grit or dust. Re-stain using a brush for smooth application and remember to rub off the excess immediately. Let dry 24 hours. Remove door and lay flat. Based on the pic, apply gloss, clear polyurethane with a foam brush to minimize bubbles. Let dry no less than 12 hours before re-hanging, preferably 24 hours.
Hope that helps ....