It's not much different from Bitcoin really. Who owns a particular coin is based entirely on the consensus of who owns what, and for a coin to change hands you just tell everyone it has.
That is in no way how modern banking works.
Only about 1/7th of the money in peoples bank accounts in the US has a physical counterpart in any bank vault, almost all of the "money" in bank accounts is purely virtual. Most money today is not printed by the mint but instead wizarded into existence in a fashion pretty much the same as how world of warcraft gold is created, but with regulatory restrictions in place around the wizardry to create desired economic outcomes.
When a crisis prompts the population to collectively attempt to withdraw more than this reserve 1/7th, it's called a "run on the bank". The bank is unable to fulfill it's obligations[1], and economies break down as faith in the currency collapses. This is exactly what happened to Cypris in 2012. The whole system basically hinges on a significant majority of the population not scrutinizing the banking sector too closely.
If you'd like to know more, there are quite a few great youtube videos on fractional reserve banking, and it's absolutely worth investing an hour to watch one of the longer ones. This one is pretty good and approachable, although it's certainly biased: https://www.youtube.com/watch?v=4AC6RSau7r8
[1] It's actually more complex than this. Central banks can help prop up other banks in this case, but central banks also have their limits in the case of a nation-wide run on multiple banks.
With fractional reserve banking the bank isn’t lending out your money... they’re lending out their own money, and you also have a contract with the bank for a similar amount of money. When a crisis like Cypress happens, it isn’t a failure of the bank to return its customers money, it’s a failure of the bank to honor their contracts.
In principle, if everyone who the bank had loaned money to was able to immediately repay it, then you could unwind everything and make everybody whole again. The bank run occurs because this can't happen - the people the bank has loaned the money to simply cannot repay immediately, because they've spent the loan on non-liquid assets.
The fundamental purpose of the bank in such a system is to fund long-term, risky and therefore expensive debt (such as to someone who wants to start a business) with short-term, secure and therefore cheap debt (eg a savings account).