Yes, but by using that definition all baseload power generation (coal, gas, hydro, nuclear, etc.) are "energy batteries", which makes the term meaningless. By "energy batteries" when talking about the electric grid, people normally mean different kinds of large scale rechargeable energy storage solutions, such as pumped storage, molten salt, etc.
With oversized generators hydro provides an extra storage that can last weeks and even months at the pick usage depending on reservoir capacity. This is the battery effect that cannot be matched with molten salt.
As I just stated, yes, that's called baseload power. It applies equally to hydro power, nuclear, coal, gas and all other baseload types. If you make them larger they can produce more. "Energy batteries" when talking about the electric grid are generally rechargeable, because as I stated otherwise the term is meaningless and just means "baseload power" in general.
Increasing capacity at hydro power stations or other baseload stations isn't as easy as just "putting a larger generator in". Everything needs to be considered and sized up: turbines, piping, outlets, new environmental assessments, etc., just as in any other type of power plant.
