Yes, but with the “revolving door” between private financial institutions and government financial policy/regulation, there’s little real distinction anymore between the two.
Those private banks can print that money out of thin air because government allows them to. And the government officials (many formerly financial executives) allow them to because they “have to” to prevent “disastrous” private banking/financial collapse.
But if you or I wanted to play the same games to print our own money they way they do? No, that would be wrong and dangerous and illegal!
So it’s pretty clear that both government and private financial institutions are tightly coupled partners in a mostly corrupt, intentionally obfuscated shell game that primarily serves to keep money and power steadily flowing into the hands of the already wealthy and powerful.
Just look at who is actually held accountable for financial crimes. Some individual trader that finds and exploits some glitch that allows them to profit from the wealthy? Straight to jail. High ranking institutional powers (government and private) that implement often illegal schemes that continuously siphon wealth from common people into their hands? Slap on the wrist at most.