Power of a capacitor when is connected to a battery
Power is not stored. Energy is stored. Power is the rate at which energy is transferred.

Think about your electricity meter. You get charged for kW hour usage. kW hours are "so many" kilo watts consumed x the number of hours you have consumed those kW for. You have a big number on your meter that tells you how many kWh you have accumulated over so many years. That number is your total energy usage over that period. 1 joule is 1 watt-second. 1000 joules is 1 kW-second and 3.6 M joules is 1 kWh.

If the number on your meter was unchanged over a short period of time then the power taken over that short period of time is zero.

In other words, energy can be stored (or metered or accumulated) over a period of time and the value it accumulates to remains steady even if the actual power drawn (at a particular moment in time) is zero.