Looks like the finish is releasing due to moisture or heavy use. The way the color comes off may indicate a lower-quality all-in-one finish, rather than a conventional penetrating stain/varnish overcoat approach.
I'd purchase a matching stain (somewhere in the mahogany or cherry range) and wipe that on. If it does well, apply semi-gloss polyurethane or acrylic varnish to the affected areas. You'll need to cover an entire board to get a reasonably good finish, so plan on masking off the inset panel and varnishing the entire frame (once you confirm that the sheen will be acceptable), or varnish the entire door.
For a really nice repair you'd need to strip or sand the old finish off, but with routed profile frames like this that can be difficult.