Watt hour is a unit of energy. Energy is the capacity to do work. When the batter leaves the factory, it is empty. It has no energy, but it can hold energy if you give it some.

It's like if you produced gallon-sized plastic water jugs. Let's say you made 1000 a day. Would it be accurate to say you produced 1000 gallons of water a day? No, you produced 1000 empty jugs.

Batteries are charged when they leave the factory. Besides that 1000 gal/d of storage capacity in your example is a valid unit for specifying factory output, is it not?

Yes, "1000 gal/d of storage capacity" is valid. "1000 gal/d" is not. Even if you used 1000 gal/d colloquially, you would have been talking about the capacity of the bottles.

The misconception you're having here is that "GWh of storage" is the volume of the bottle. You're interpreting it as "GWh", which in the bottle example would be like saying "They produce 1000 gallons of water / d".