One of the plates is anodized to a certain voltage, and the other either a lot less, or not at all (native aluminum oxide).
Anodization is diffusion mediated as far as I know. You can break it down to an atomic description where Al + 3H2O --> 1.5 H2 + Al(OH)3 (H2 being produced at the cathode, Al(OH)3 at the anode, as this is an externally powered, electrochemical process), but the structure develops due to the atoms and charges diffusing through the bulk material, and all that's required is a model of that bulk material.
Thickness of the [hydr]oxide layer determines voltage rating.
Probably there's also a free 1.2V or so, due to the electric double layer effect (limited by the reaction voltage of the electrodes and decomposition of the electrolyte). When an electrolyte is used that is less prone to decomposition, this effect can be harnessed to better effect. EDL capacitors (supercapacitors) use organic electrolytes like ethyl carbonate, which decompose above 2.7V, hence the 2.7V rating of single-cell supercapacitors. (Combined with an extremely high surface area electrode like activated charcoal, this gives usefully large capacitances, in the farads. Obviously, electrolytic capacitors don't have as much surface area, but they have higher voltage ratings too.)
Tim