My guess would be that solvents in the cleaner/polish dissolved the synthetic material the vanity is made of.
Scrub aggressively with a dry cloth towel to ensure that the problem isn't just residue left behind by the cleaner/polish.
If scrubbing with a cloth doesn't help, then my advice would be to put a hard acrylic film over the counter top. For example, Flecto makes an acrylic product that crosslinks to form a film hard enough to stand up on a floor:
You don't need to know the rest:
The reason why your counter top will look glossy when it's wet is because of "Fresnel's Law". Fresnel's Law says that the amount of light reflected from an interface will be proportional to the ratio of the difference in refractive indices across that interface over the sum of the refractive indices all squared.
Despite the fact that both a clear acrylic finish and air are both transparent, the fact that the acrylic finish is a solid while air is a gas means that the acrylic finish is gonna have a refractive index much closer to other solids, like the solid synthetic marble of your vanity.
Thus, according to Fresnel's law, if you coat your vanity with a clear film, then MUCH more of the light reflected from that surface will be at the air/film boundary and much less of it will be at the film/marble boundary.
And, that's simply because you'll have a much greater difference in the refractive index at the air/film boundary than the film/marble boundary. Thus, the difference in refractive indices across the air/film boundary will be large, making the amount of light reflected from that interface large. Similarily, the difference in refractive indices at the film/marble boundary will be small, making the amount of light reflected from that interface small. Since you see the light reflected from the vanity, you will "see" a smooth surface on that vanity.
So, while others would have told you to put a clear coating over your vanity, now you know why a clear coating on the vanity will restore it's smooth, glossy appearance.
Yep, I'm not too busy today.