Being too lazy (and busy with other things) to try to derive the
relationship for sure, intuition tells me the force distribution on the
screws is as follows:

Let's say you have three screws per stud and you place them at 3", 5",
and 7" from the bottom. I think the force distribution is 3/(3+5+7) =
20% of the total force on the bottom screw, 5/(3+5+7) = 33% on the
middle screw, and 7/(3+5+7) = 47% on the top screw.

I think (anyone else please tell me if you agree or disagree), the
total force referred to above is still the same as if there was only
the top screw. Adding the other two screws essentially cuts the force
on the top screw in half by absorbing the other half between the two of
them.

By the way, we're referring to the outward (tensile) force on the
screws caused by the torque. There's also a downward (shear) force
divided equally among them which is simply the total weight of the
vanity and anyone sitting one it divided by the total number of screws.
That force shouldn't be much of an issue.

Lastly, regardless of how many studs you screw into, there will still
only be two wood arms that reach out to hold up the vanity. Assuming
that the arms are held onto the backer board with screws from the back
side, the force calculation on those screws is just the same as before,
except that the total force will only be divided between the two arms.
Having three or three hundred studs to screw the backer board into
doesn't help for those screws.