There's some damage/discolouration to the wood around the original tap hole
Depending on your skill/comfort level, and the value placed on this benchtop, you could patch with a similar or contrasting piece of wood. However nice clean end-grain cuts are hard enough, and more challenging in old wood.
To blend a repair section with the original, you could re-finish the entire wooden top (strip, apply wood-filler, sand, sand, sand some more, stain and then seal with clear top coats) If you do this, I'd suggest a whole strip across the back of the sink, wider than the two tap bases and all the way back under the upstand, and to make it even.
Aside: we can see where the planks have been joined to make the top surface in the first place. You might be able to find donor wood from elsewhere on the unit, that can itself be replaced with something less visible.
A slightly easier solution might be to add a square tile under each fitment, or tile the whole strip under 3 fittings. There are tiles intended to butt up against each other - you don't want that, you'd want tiles with rounded edges, and to seal them well.
The cheap solution is to put a backer on the underside, and fill the hole with builder's bog/fill to just below the surface, and then use colour-matched wood putty. And seal really well. Downside of this is that it will never look "right"
...however you could be arty, and carve/route a design into the surface such that it incorporates the two holes. Fill the channel with a contrasting colour like blue or red that matches the room, and seal it well. Kind of like the furniture-equivalent of drawing over a bad tattoo. Additional decorative motifs could be added elsewhere on the top/front to draw the eye away.
Ultimately it depends on the value-to-you of this unit, and your time.
The lazy solution is a couple of well-placed cups, one for toothbrushes, and a can of shaving cream, or whatever you routinely have on your bathroom bench. Small potplants or similar perhaps.