Wood stain comes in two broad categories, dye-based and pigment-based. The oil-based stain you used is the latter type, either entirely or mostly coloured with pigment1 suspended in an oily binder. Pigment is actual granular particles and even though the particles are very tiny it doesn't soak into the wood like dyes do. This is why oil-based stains are known to obscure grain more than water- or alcohol-based stains, they actually sit on the surface more than being absorbed.
Since this was done with oil-based stain you might be able to remove some of it from the surface by wiping it away with its original solvent, mineral spirits (UK: white spirit). Just like with oil-based varnishes if the curing process hasn't progressed too far it will still dissolve in its original solvent, but if it has cured too much it'll work weakly or not at all.
The usual next thing to try if spirits don't work is a stronger solvent, acetone being the usual pick. But I think here it would probably be best not to try that because the substrate is MDF and not solid wood. Better then to paint and possibly then varnish, or scrape/sand back to clean wood and begin again with your original plan.
What now?
Back to bare wood
This is the usual coarse of action when a stain job has gone wrong. But partly because this is MDF and partly because it's a sewing table I think you should give thought to painting instead.
Firstly because it will save you from all the MDF dust. Due to the structure of MDF (it's wood fibre bonded together with resin) scraping doesn't offer the usual advantage that it does on solid wood, where you mostly produce small shavings rather than dust and leave a beautifully smooth surface behind.
Scraping still works on MDF, it can remove the surface very efficiently, but once you've gone through the factory face the material immediately under is noticeably more fibrous and open-textured, so the surface won't be pristine and beautiful and it is usually a bit of a pig to refinish2.
Paint
You can paint directly on top of oil-based stains once they are fully 'dry'. Oil-based interior paints for trimwork are usually alkyd enamels these days and they're quite durable and would be a good pick here.
Painting saves you the time and effort of scraping/sanding and you won't have lots of dust to deal with. A light scuff-sand or rub over with Scotch-Brite or steel wool to prep the surface and you can start painting. Apply with a foam roller for best results.
For maximum wear resistance you can varnish after you paint3, although many oil-based paints are acceptably tough and durable. Even though the surface may not be quite as tough unvarnished there is an advantage down the line, if you get wear-through at edges or corners you can easily repaint those areas. Touch ups like this would be more difficult to blend in if you'd varnished.
Flip it over and use the other side
Not sure what if anything you've done to the opposite side that would prevent you doing this but if you've done nothing to the MDF it should be just as it came from the store and you can start again with no further effort expended.
This may be so obvious you didn't think of it, it's surprising how often this is overlooked as an option!
Edges
In case you weren't planning on it, I'd recommend you either apply solid-wood edging strips to protect the edges of the MDF (even 1/8" material is worth applying), or put in a small chamfer or roundover on the MDF which will improve its wear characteristics.
1 Some pigmented stains have some dye in them as well.
2 It's not as smooth and it's absorbent like a sponge (very like the edges of MDF).
3 Regardless of paint type. Waterbased wall paints (i.e. "latex" paint in the US, emulsion paint in the UK), oil paint, enamels, acrylic enamels, milk paint and chalk paint can all be varnished although in the last case it would be a complete waste of the original paint.
