I will probably just show my ignorance here . . . but here goes.
My grand piano is glossy black, using a polyurethane varnish. This type of varnish is essentially a liquid plastic. I'm sure it can be scratched, but a lightly moistened cloth should not hurt the finish. Nor should it be able to penetrate the finish to the wood.
There's a big difference with pianos that have a velvet finish, or utilize traditional varnishes. These would be more vulnerable to moisture.
I have committed heresy here on the forum in the past, stating that I use lemon Pledge to clean my piano. I used to use a very expensive finish incorporating a wax, called Plush. Unfortunately, after quite a few years, I noted much cloudiness, even though I tried to rub out the Plush when I applied it. Turns up to be old wax build-up that was oxidizing and looked awful.
The Pledge was a product that cuts the wax buildup. Using Pledge and much elbow grease, I did manage to get my piano back to her old glossy self. It took time, lots of work, and multiple attempts. Because Pledge has a volatile base (some sort of volatile distillate) it does not and cannot build up like waxes. It also cuts old wax rather effectively.
One poster responded to me that no trace of Pledge should ever reach the pinblock or it will loosen the pins. So I make a real effort to avoid getting the product in such a place. Actually, I spray a little pledge on a dusting cloth, and never spray it on the piano itself.
If persons DO respond to this and tell me I should not use Pledge, I hope they do not recommend that I go back to a wax type product.
Best to all--