Are you saying that you can't measure battery power capacity? Why would you assert something weird like that?
Grid assets are measured by power, and power is a key specification of storage capacity.
Edit in response to your edit: the EIA, and in fact all the people who run grids and the grid storage, refer to capacity in terms of power, and when talking about "storage capacity" on the grid continues to use power because it would be nonsensical and silly to switch to a not-so-useful energy unit rather than power unit when power is the key metric for grids.
That seems of limited use without knowing for how long the power can be supplied. 1GW for 1 second is pretty useless. 1GW for a minute not much better.
Most people would say that 1GW lasting a day is more storage capacity than 10GW lasting only an hour.
Wouldn't batteries optimized for power take the form of supercapacitors [1] instead?
"Seconds" of capacity might be useful for "primary frequency response" but not for any of the other functions listed in table 1 of https://www.nrel.gov/docs/fy19osti/74426.pdf. Primary frequency response is only seen as important in a limited number of markets in the US (see that table).
It turns out that the resources that get deployed in the grid are useful ones rather than something that deploys 1GW for 1 second. So referring to the power capacity, and dispatching based on power capacity, is how to make decisions when operators bus into the grid. At any moment, a power source may trip off, or decide that the price they are being paid is not high enough to justify continued operation.
The grid is an amazing, absolutely massive, machine for matching power at the generation and load sides. Power, of both the real sort and the reactive sort, are the name of the game.
As far as optimization for power over minimal capacity, the best use case for supercapacitors might be frequency regulation, but batteries have proven to be superior for that particular use case, with a few fly wheels also taking part.
Grid assets are measured by power, and power is a key specification of storage capacity.
Edit in response to your edit: the EIA, and in fact all the people who run grids and the grid storage, refer to capacity in terms of power, and when talking about "storage capacity" on the grid continues to use power because it would be nonsensical and silly to switch to a not-so-useful energy unit rather than power unit when power is the key metric for grids.