Series and parallel Lipo question (20,000mAh capacity)
What batteries actually store

Batteries don't store charge, they store energy. That energy can be used to move charge. When a battery is being discharged it is receiving as much electrical charge at one terminal as it is emitting at the other terminal. There's no net loss or gain of electrical charge.

The energy stored is V * I * t (though you really need to integrate  since voltage will be a function of time as the battery is discharged)

The numbers in your question are equivalent:


3.7 * 20,000 = 74,000 units of energy†
14.8 * 5,000 = 74,000 units of energy




Battery Capacity


  A battery's capacity is the amount of electric charge it can deliver at the rated voltage. 


Charge, as we know, is measured in coulombs, an amp is a coulomb per second, so you can also express charge in amp-seconds (or amp-hours or milliamp-hours etc)

If you reconfigure the cells of the battery so that the battery has a higher rated voltage, the capacity (by this definition) is correspondingly reduced (since the overall energy available must be the same as before reconfiguration). 

The energy is V * I * t so if you increase V you must decrease I * t

(which is what battery makers call the capacity - measured in amp-seconds although they usually apply a constant so they can express it in amp-hours or milliamp-hours for the convenience of consumers)



The claim in question

Lets look at the claim you seem to be disputing 


  "the power is the same". How can that be?


It's just sloppy language. They are trying to say that 20,000 mAh * 3.7 V delivers the same energy as 5,000 mAH * 14.8V - which is correct.

You can of course increase power by increasing V or I - however you thereby decrease t in inverse proportion.



Footnotes

† Units of energy

1 watt = 1 joule per second
   1 watt x 1 second = 1 joule
   V * I = watts
   1 volt * 1 amp * 1 second = 1 joule

The units in my answer are not joules (since the times are hours not seconds and the currents are milliamps not amps) but that's just a scaling factor that is the same for both numbers.