When a banker records how many dollars each of his borrowers owes the bank, money is serving as a:?

When a banker records how many dollars each of his borrowers owes the bank, money is serving as a unit of account.

Unit of account is the actual money, or a sum that something costs. So here, the amount of money that the borrower needs to return to the bank is the amount that his borrowing costs, which is why it is called a unit of account.