Or some spirit stain (looks like Rosewood in the photo) dissolved in meths. Mix it weak and build up the colour with multiple coats. The problem is two fold, matching the colour and matching the gloss/sheen, fortunately on an edge like that you can get away with murder in terms of gloss differentials.
Just to clarify, the original stain/finish looks like Rosewood, the wood is almost certainly Beech. Beech is the traditional choice for framing soft furnishings.
If I ever had to repair anything like that or make anything from a light wood that would have a dark finish I always tried to dye the wood first, before applying the coloured finish - if it did rub through it was then a bit less conspicuous.