Why does old gasoline seem to have more "kick"?
My candidate to explain the increased performance is the removal of water as part of the corrosion reaction occurring in metal storage containers. To quote a reference:


  Since oxygen has different solubilities (higher for the organic and lower for the aqueous phase), anode or cathode regions are formed on the steel or metal surface. This creates favorable conditions for electrochemical corrosion in a given environment.


As such, I would argue that not only does oxygen content impact corrosion rate, but potentially more important, is that the removal of water over time (from corrosion chemistry) also increases O2 solubility (per above). This could be especially favorable when the fuel is vaporized and mixed with air in the ignition chamber.

[EDIT] Further comments, same source:


  Oxidation of biodiesel leads to the formation of various products such as peroxides and hydroperoxides. During degradation, these products transform into shorter-chain compounds such as low-molecular weight acids, aldehydes, ketones, and alcohols [9].


Per another source:


  In the winter, companies produce a gasoline containing lighter hydrocarbons, making the liquid more volatile and therefore easier to ignite.


This source also notes with more volatile fuels, evaporation can leave behind a heavier fuel mix.

Corrosion induced oxidation of biodiesel, in particular, appears to not only impact potential oxygen content of the fuel, but may add peroxides as well. The creation of lower molecular weight organics should also increase flammability.

An interesting point once presented in Wikipedia on Methanol, to quote:

"As a fuel for mud racers, methanol mixed with gasoline and nitrous oxide produces more power than gasoline and nitrous oxide alone."

Which, in the current context, supports my opinion that fuel mix chemistry (whether corrosion based, or otherwise, involving O2 or N2O) CAN IMPACT PERFORMANCE.

So, more oxygen content, less heavy fuel mix, more complete ignition, may explain some of the so-called 'kick' that has been referenced. On the other hand, significant evaporation is likely detrimental.

[EDIT] Yet another path as a consequence of corrosion chemistry, to quote from Wikipedia on Tetraethyllead:


  At the temperatures found in internal combustion engines, (CH3CH2)4Pb decomposes completely into lead and lead oxides as well as combustible, short-lived ethyl radicals. Lead and lead oxide scavenge radical intermediates in combustion reactions. Engine knock is caused by a cool flame, an oscillating low-temperature combustion reaction that occurs before the proper, hot ignition. Lead quenches the pyrolysed radicals and thus kills the radical chain reaction that would sustain a cool flame, preventing it from disturbing the smooth ignition of the hot flame front.


Also on the performance effect of Tetraethyllead (TEL):


  Antiknock agents allow the use of higher compression ratios for greater efficiency[17] and peak power.[18] Adding varying amounts of additives such as low percentage TEL or high percentage ethanol to gasoline allowed easy inexpensive control of octane ratings... Aviation spirits with TEL used in WWII reached 150 octane to enable supercharged engines such as the Rolls-Royce Merlin and Griffon to reach high horsepower ratings at altitude.[19] In military aviation, TEL manipulation allowed a range of different fuels to be tailored for particular flight conditions.


So, the corrosion process on even minor alloy metals, found in the metal storage container, may have also introduced an organometal compound with similar properties to TEL in its ability to scavenge radical intermediates.