Question

Why is the proportionality of charge and the potential of it's surface independent of the shape of the conductor?

Answer 1

Basically that’s how capacitors work. Your question is like asking why when you pour water into a bucket does it get more full.
An ideal capacitor is wholly characterized by a constant capacitance C, defined as the ratio of charge ±Q on each conductor to the voltage V between them. Rewritten and reordered,
C=Q/V
and Q= C x V.
You should be able to see how charge and voltage are proportional.
and so it does not depend on the area of the conductor. And hence shape doesn't matter for this proportionality.