Although battery can store energy from any source, the production process may make a different storage beneficial.
Electronic production like photovoltaic may be best stored in a battery.
Hydro production is best stored in a reservoir.
Nuclear may be best stored as heat in a molten salt.
Intermittent wind energy is best converted to compressed air at a low cost for storage. It could be used via pneumatic route or converted to electric at scale of useage.
The only reason we are discussing electric storage is because wind and solar energy must be harvested as it is produced. Nuclear, hydro, coal, geothermal, and so on is already "stored" by not converting it to electricity in the first place. If we don't burn coal then it's not lost to us like if we don't allow a windmill to spin in the wind.
This problem of intermittency of wind and solar power is not near as trivial as many would imply it to be. This is on top of the current costs of these energy sources. Producing a viable electricity storage system is required for wind and solar to meet more than a small fraction of our energy needs. If we also do not see a massive drop in the cost of wind and solar energy then not even a free battery can make it work, and nothing is free.
If we simply ignore wind and solar energy, which I believe we should, and move to next generation nuclear then we won't need these utility scale batteries. These batteries might still be useful for things like keeping data centers running in emergencies or something but that is much smaller scale than what we'd need for wind and solar to make a dent in our energy needs.
Lacking next generation nuclear and adding some sort of utility scale storage then wind and solar look kind of pointless as an energy source. They look rather pointless now. Again, wind and solar require storage but that is insufficient, they also need to come down in costs. I've seen Prof. Sadoway on a TED Talk video and he certainly makes a good case but this all evaporates once the costs of wind and solar are taken into account. Nuclear, coal, hydro, etc. do not require batteries by their very nature. If we can get these batteries produced then they can possibly make these already cheap energy sources cheaper still. That makes for an even higher bar for wind and solar to clear.
Disclaimer: I am an engineer but not a nuclear engineer, mechanical engineer, chemical engineer, or industrial engineer. My education included electrical, computer, and software engineering.