If the surface is edge grain and the previous finish was only wax, wiping down with mineral spirits and then sanding using progressively finer grits up to 220 or so should be pretty good preparation to refinish. Skip the soap and water treatment as it won't dissolve any wax and only complicates finishing. Before applying any finish the wood should be smooth and dry. Any staining or blotching you can see is not going to be hidden too much by a clear finish - it's sometimes possible to bleach out stains but this is tricky to do well.
If you are still getting something soaking back up after sanding down to the raw wood, it could be a prior mineral oil treatment. Mineral oil never hardens and is often used on wooden butcher blocks and countertops. Simply reapplying more would be one refinishing option. It's easy (you just apply until it is no longer absorbed) but you have to reapply it frequently. Inexpensive pure mineral oil can be found in a pharmacy (it's sold as a laxative).
Another option would be a pure tung oil finish. It would be slightly more expensive but a little more water resistant. You can get it from a woodworking or finishing supplier. Most "tung oil finishes" sold by hardware stores are not actually tung oil so check the labels carefully if you go this route. It would be applied and reapplied similarly to mineral oil.
Some "Danish oil" finishes are a blend of varnish, oil, and thinner, and others contain only thinned varnish. Generally one applies "Danish oil" in light coats such that a film is not really built up on the surface of the wood. Often wax is then applied over it. This feels nice on furniture but it is going to be less durable that a slightly thicker application of regular varnish applied with a brush or thinned and then wiped on in several coats.
The mineral oil or tung oil would be my choices for a counter that you are actually going to cut on. On the other hand if you don't plan to cut with knives directly on the counter, and if you are able to prepare the counter down to a clean, dry, smooth surface, a wiping varnish is probably the most durable and best looking option.
A few things here. First, you need to remove ALL the mold. Second, you need to prevent it from coming back. Third, I would fix that grout in the tile on the corner to prevent splashes from getting in and causing mold in there too.
You're going to need to remove the faucet completely. The hole that that thing is set in is most likely a mold farm. Just shut off the water, unhook the hoses and undo the nut clamping it to the bottom. The tap should come out the top no problem. Wear a mask, I'm expecting the worst for that one.
Now you're going to have to clean that mold, all of it. If it's caked on, chisel it off with a screwdriver or something, then get some sandpaper and make sure it's all gone. Check under the countertop too, there might be some down there. If it was me, I'd take the whole sink out and inspect, but that's probably overkill. However, if the caulking around the sink isn't very good, do go head and disconnect the drain pipes (there's a flange that unscrews) and pop the sink out.
Now, we're all clean, good. We have to fix the color. I'd get some very fine sandpaper and disregard the finish of the wood, it's already going to need repairs. Clean the mold off with the sandpaper, then get a fresh piece of sandpaper, sand down a bit more, then a really really fine grit sandpaper and do a final sanding. With the really really fine stuff, you might be able to feather the edge of the stain. If not, that's fine. You should have a good area of plain wood to work with, and the faucet should still be out.
Now, I'm not sure if that countertop is stained or if it's oiled. It's most likely oiled, and if that's the case, go ahead and do some research on how to replace the layer of oil. If you want, perhaps explore doing a layer of urethane, that will prevent mold very well, but I don't know about the aesthetics.
Anyways, now we have no mold and a fresh canvas to work on. If you took the sink out, make sure to caulk the edges very well when putting it back in. When you put the faucet back in, put a big fat bead around the hole before you replace it. Quickly tighten the nut a bit below, then get above and clean the edge of the caulk off with your finger. This will seal it so you won't have any more mold getting in the crack.
At that point, just make sure to keep the area dry to prevent future mold, or have a preventative coating on the wood. You could always bring the tile out a bit too, just for that square area of the countertop. If you decide to do that, make sure to install the tiles properly and to do it before you do put the faucet back in.
Best Answer
Colored wax is very minimally disturbing (to the existing finish) and easily reversible, should you want to redo the repair. Buff it with soft cloth to match sheens and you can even overcoat the whole piece with a good paste wax, afterwards.
For wood grains that have a range of colors, I like to blend (rub in) several colors along the crack. That way, the repair is less monochromatic and therefore more natural looking.