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.
I'm assuming this was a standard oil finish like a Danish oil or tung oil. They will harden. If the oil finish was a mineral oil, commonly used for butcher block tops for example, that oil will not harden. But even so, one coat of oil finish is not that much of a finish.
You should be able to apply stain to it. In fact, a hardened oil finish will help that stain so it is applied fairly evenly, although it may not soak in as deeply as it might otherwise. There are different kinds of stain, and that will be a factor. Thus there are dye based stains, which get into the wood. With an oil finish, the dye will not penetrate as well. Pigmented stains (essentially thinned paints) have pigment in them. (Ok, maybe that is obvious.) They are designed to sit on the surface of the wood anyway.
As far as a varnish finish, yes, a hardened oil finish that has soaked into the wood will not do anything to stop a varnish from hardening. However, if the oil finish is a non-hardening mineral oil, then varnish will probably not adhere terribly well to the wood. So really it matters what kind of oil finish was applied, and what is the purpose of this top.
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A 4-inch wide razor scraper might remove much of the dried oil. Hold it lightly and at a small angle to the surface. But you will have to be careful to avoid letting a corner of the scraper dig into the wood.