At first, we can simplify the specific capacitance into units of current and time, as shown in the equation below, with voltage present in the denominator part of the equation.
C / A = q / V.A = I.t / V.A units: [A.s/V.cm2] = [mA.h / 3.6 . V . cm2]
Therefore, If we multiply the specific capacitance by potential (V) we have the following equation, that may represent the specific capacity (lets name it as X here):
X = C.V / A = q.V / V.A = I.t/A = q / A units: [A.s / cm2] = [mA.h / 3.6 cm2]
With the units you were looking for. Therefore, the specific capacity represents the stored charges accumulated in the capacitor's plates per unit of area.