Banks certainly don't “own” the money deposited, which is why we fret about bank runs and bankruptcies.
When you make a deposit at a savings account, you've agreed to lend the money to the bank in exchange for an interest payment, however small that may be. The bank will then use your money to earn their spread, which is the difference between the interest rate they pay you and the interest rate they earn from the next borrower. Borrow at 0.01% and lend at 4.25%, for example.
It's a pretty sweet scheme if you think about it. A bank is legally licensed to make money with your money and the more you deposit the better. That why banks are always offering incentives for additional deposits to savings accounts. And, this helps to explain why banks are able to afford the nicest buildings of any business.
Checking accounts are another story as banks are required to keep those funds liquid for you to draw on at any time. They can shuffle around funds to meet your withdrawals, but they can't put all of it into higher interest yielding loans.