Wow, thanks for the extra detail on the various grains and the advise on possible alternatives. e.g. the legs being different colour from the base. This is a last resort but might end up the pragmatic solution. We like the darker grain detail on the top so that won't be a problem - it is the legs that just look patchy. Anyway, let's see how the Oxalic acid works. The whole purpose of the exercise is to get away from the dark finish, hence striving to solve the problem rather than cover it up. Thanks again for the level of detail, much appreciated.As the great waldo pointed out the table is made of two different materials; the top is coarse textured ring porous oak and the underframe is fine textured diffuse porous beech. There are remnants of the old pigmented varnish lodged in the pores of both wood species. In the oak it is mostly apparent lodged in the distinct dark bands of the coarser and more open spring growth. In the case of the beech which has a fine grain lacking a marked distinction between spring and summer growth density and texture the remaining pigmented finish is relatively evenly spread throughout the small pores of the timber's surface.
I suspect the bleaching options suggested by others, whether you use A+B bleach or oxalic acid, or both, won't remove the black colour from both the wood species. If either or both of the bleaches work then you could follow up with a dye of your choice and refinish with varnish or similar. If, on the other hand, the bleaching strategies don't work you might consider applying different finishes to the different parts. You could, for example, paint or dye/stain plus polish the underframe one colour and finish the oak top just with a clear finish, such as an oil varnish which itself will darken the oak, but leave the dark pigmented bands of coarse spring growth.
I don't think you'll ever be able to get the underframe to closely match texturally the appearance of the oak because the two wood species have different texture (coarse and smooth or fine), but you should be able to get a close colour match by finishing both with a dark stain or dye plus polish. However, that is the look you've just removed and you want to get away from that, so maybe the two different treatments for the two wood species you're dealing with might be worth considering. Slainte.