The only time that "storing money" would make sense would be if you were hiding it someplace, and even then "hiding" it would sound more natural. "Saving money" is a natural and common collocation, and "storing" it just simply isn't. As I often say when answering these types of questions here, you should always strive to learn the collocations that natives use rather than try to invent your own new ones.

STORING MONEY only makes sense if you are a bank, or other institution that has so much money you have to figure out where to put it. Or it might refer to a business keeping its cash in a safe. For your context, SAVE is the word you want.

Yes "store" is correct and it is a good word to use. It is not the word that most people use, but who cares? Is it "common usage"? It depends whom you ask. Financial types tend not to use "save". "Save" is used more by the little people, meaning most of us. "Park" is a word I like to use, as in "let's park this in a safe place".

to store money (away) = to stash money (away.) (hide money)

It's not "idiomatic." "Storing" is just not a word that goes with "money." As with many things in language, I can't give a logical explanation. Unfortunately, language is more than vocabulary + grammar. There is also "customary usage." "To store" is often used with words for physical things. People will ask about "how to store rare coins" or "how to store gold." It is is used with words for some non-physical things: a hard drive "stores data." In old-fashioned language, we read about "storing up treasure." But it isn't used with "money." In technical discussions of economics, "money" is defined as something that serves three function, and one of them is "to serve as a store of value." However, "store of value" isn't heard in any other context.