To me the fundamentals have always been based on batteries being chemical reactions with their characteristic cell voltages. A cell definitely has its own discharge voltage, and can be used as a reference because the voltage remains relatively unchanged until the available chemicals are becoming effectively exhausted. It is an energy source based on the amount of chemicals available, plus a storage medium based on the reliable reversibility of the reaction.

In the case of rechargeable batteries, the desired reverse reaction can often be best obtained by carefully matching the recharge voltage & current dynamically, against the quantity of chemical you wish to react, including the amount undischarged and remaining to be reacted at any time. The recharge/maintenance voltages will need to be above the discharged/characterisitc voltages any time that cell recharging is expected to occur. The upper limit on charging voltage is based on the cell's characteristic voltage & the ability of the particular chemical system to withstand overvoltage. This is an electrochemical limit. The electrodes are separated by an electrolyte where having a low DC resistance is a desired property. As to the polarity of the DC voltage being provided or stored, this should always be respected, at your peril.

The working voltage of a battery cell can not be adjusted to match a particular circuit, the circuits must instead be closely designed to match the cell's characteristic voltage and chemical behavior. Increased energy storage requires more voluminous chemicals at the same voltage.

A capacitor whose plates can store charge across a dielectric has no characteristic voltage.

A capacitor can be a power store but there's no electrochemistry required.

Semantically, you can not "recharge" a capacitor since there is no original charge to begin with. You must first charge it with an energy source, which a battery by definition contains but a capacitor does not. A capacitor stores energy but it is not an actual energy source.

When charging a simple solid-state capacitor the upper limit on charging voltage is based on the electrical resistance of the particular dielectric & its ability to withstand overvoltage. This is a physical limit. The electrodes are separated by a dielectric where having a high DC resistance is a desired property. A simple solid-state capacitor functions equally well for operation at reversed polarity.

For an electrolytic capacitor, it is polarized because the chemicals it contains provide an effective increase in storage per volume, at the disadvantage of having high DC resistance in only one direction. An additional limit in maximum voltage is imposed by the electrochemical nature of these type capacitors, but they are still rated as against a physical limit. Plus the polarity of the DC voltage being stored in them should always be respected, your peril is once again at stake.

But no capacitor has its own ideal voltage, they are not at all like batteries in this regard.

A battery's maximum charging voltage is usually limited to a range close to its nominal voltage, imposed by the natural scientific ability of the chemicals to withstand overvoltage while retaining composure. There is very little leeway to work with here. To change the working voltage of a battery storage bank requires addressing the granularity of a different number of matched cells in series, plus circuitry carefully re-optimized for the incremented working battery voltage.

A capacitor's maximum charging voltage is only limited to a range within its rated voltage,
which is an engineering value assigned based on the natural scientific ability of the dielectric to withstand overvoltage while retaining composure. The order(s) of magnitude more leeway should be easily recognized. To change the working voltage of a capacitor storage bank requires only addressing the dielectric strength, which would not need to be changed if it was over-specified to begin with.

The working voltage of capacitor storage can be any arbitrary voltage selected for various engineering reasons, the working voltage only needs to be selected below the electrical rating of the components. Increased energy storage requires more voluminous dielectric, whether more capacitors having the same voltage rating, or bigger packages handling higher chosen voltages.

One of the differences between electronics and electrochemistry, capacitors are nominally marked with a usually conservative rating, but some parts may often be capable of truly handling twice the rated voltage reliably. Batteries not so much.

As a natural scientist designing circuits primarily using natural intelligence through experimentation & discovery, key efforts in computer science seem to be 100% helpful when invested in calculations which are too heavy or numerous otherwise.

As a computer scientist designing circuits primarily using software simulation, when key efforts in physical prototype building fail, this might reveal the investment in computer science itself to be less than 100% helpful.

Don't get me started on electrochemistry . . . the potential for reaction could be unlimited ;-)