Common name for institutes similar to banks or building societies that allow you to save or deposit your money is – Depository
Answer : Depository
The word Repository is also a good candidate but find out below why it isn’t the most appropriate word:
Depository VS Repository:
A Depository is a place where you deposit something and can get the same value in return but not the exact object. For example, you may deposit 100 dollars at a bank. You have the right to get it back whenever you want (depending on your agreement, of course) but you will not get the same Notes or Coins you deposited and you may even get it in different amounts (e.g. five withdrawals of 20 dollars instead of a single 100 dollar withdrawal).
Whereas a Repository is a place where you store things like a warehouse. You will get the Exact item back (e.g. If you store a diamand necklace in a secure repository, you will be disapointed if you don’t get the exact necklace back even if they offer something else with the exact money vaule).