Why are batteries measured in electrical charge unit (ampere-hour) and not in energy unit?
You may have missed tyblu's clarification on his answer. As stated, a battery can store a fairly constant amount of charge, this decreases as the battery is discharged, a batteries output energy is dependant on it's operating conditions, and partially the length of time the battery spends powering something.

This can be seen on the datasheet of many batteries. Such as this. Note that the scale is not linear on all of the graphs.

(I'm completely rusty, and just getting back into electronics, so please feel free to point out any corrections needed, or the possible non-sensical answer.)