An 'account' within the records of a bank or any other business is part of a system of double-entry bookkeeping, originated by the Venetians.
The beauty of a double-entry system, which is used almost by every business everywhere in the modern world, is that it carries an inherent check for errors. At any one time all debit balances must equal all credit balances. If they do not, then there has been some sort of error, which needs investigation.
If all debits equal all credits the books are said to 'balance'.
So why is it that an individual account has a 'balance'? Because if you add all the debits to that account and subtract all the credits, the resulting figure, debit or credit, is the one that is needed to 'balance' one side with the other.
Nowadays it is all made easy by computers doing the legwork. But anyone, like me, who began as a trainee accountant in 1961 will remember what it was like having to 'balance' a set of books manually. It could be positively mind-numbingly infuriating if they didn't agree and could involve a lot of sweat, midnight-hours, black coffee, and for some (not me) cigarettes, to find the error(s). 'Balance' was a word at the centre of one's life.