Though the sink may be soapstone, it's quite possibly slate.
Slate is native to upstate NY, and the most durable slate comes from NY and NE quarries. It is not unusual for homes in the area have slate floors, counters and sinks.

If you are anywhere near Middle Granville, NY, I would recommend a visit to the Sheldon Slate Co. We drove three hours to get there, and it was well worth it. We were given a most informative tour of the facility, and got see kitchen and bathroom displays using slate.

Here is a link that might be useful: Sheldon Slate Company

I'd guess that your sink is probably cast iron, which could also restore nicely.

Soapstone will attract oil and repel water. If you put a drop of water on it and it just kind of mounds where it is, instead of soaking in, it's probably soapstone. If the opposite happens with oil, you put a drop on and it makes a blotch, clinging to the stone, it's probably soapstone. If both things happen at once, it's almost definitely soapstone.

If there's a worn, abraded place that has dust that feels powdery, like talc, which is a component of soapstone, and it passes the oil and water tests, no matter what it is, you'd be 100% right to call it soapstone.

Some of the hardest varieties of soapstone have more besides steatite and talc, and the occasion vein of quartz in them, and aren't as oliophilic, so don't really show much during the oil test. Conversely, if your stone has built up a layer of polymerized oil (which is what people are trying to achieve when they oil their counters), the water test also might not show big results. If you scrub an area with Dawn liquid dish detergent or another mild grease fighter and a soft brush or scrubbie, you could remove the polymerized layer to know for sure. Before you bother, you could look at the underside, which probably has portions that haven't built up an oil layer over time.

Of course, those are general instructions for how to tell if it's soapstone and that implies that there are unpainted portions that are accessible. I don't know how you know what's under the paint.

When I lived in western Mass I had a slate sink in the basement. They seemed pretty common in the houses we looked at.
Assuming the paint would clean up and you have no need for it in either the kitchen or the basement, you can let it sit for the next owner of the house to discover. Or if you had a use for it outside (gardening, potting), you might get some hefty people to move it out there. Otherwise, Craigslist.

Thanks so much! You all are so knowledgeable.

The slate company linked above is 5.5 hours from me (I'm in western NY). Slate sounds like a reasonable guess, though. Maybe I'll tinker around with it a bit tomorrow and see if I can figure anything out. Thanks for the help!