Worse than that: if you buy hydrogen today (e.g. for some fuel cell car prototype), it is almost certainly created from fossil hydrocarbons. So hydrogen is not

Worse than that: if you buy hydrogen today (e.g. for some fuel cell car prototype), it is almost certainly created from fossil hydrocarbons. So hydrogen is not even a battery now but just a very elaborate way to burn fossils. "Equivalent to batteries" is the best case future scenario (e.g. in presence of abundant electricity from controlled fusion power plants).