Of course it's real money. The entire global economy is built on this.
They can't run out of money if they create it ex nihilo. They can only run out of borrowers
Indeed, it is real money. Money is "real money" if enough people think of it as "real money".
A bank account with a big number on it is seen, by most people, as "real money" so it IS real money.
You are willing to hand me over your house if I give you my bank account with a very big number on it.
That is what proves that it is "real" money.
The funny property of money is that money is money because enough people say/think so.
However, it is not because something is "real money" that it is WORTH A LOT. The "price" of money depends essentially on how it is issued, how much is circulating, how much is (expected) to be circulating, and how confident people are in its ability to be a store of value (and are storing value in it for that reason).
If "real money" is too volatile, it will stop being considered as money by many, and hence it will stop being money.
(that's called hyperinflation).