Parent and GP are using different definitions of "dollar."
GP is using a strict definition (only dollar-denominated liability of the Fed is a "true" dollar), parent is using a looser definition (a dollar-denominated liability of any bank is a dollar).
If you have a fractional reserve (e.g., a bank has $100M cash backing $100M deposits one day, then loans out $30M the next day), with the strict definition you still have $100M dollars ($70M controlled by the bank, $30M that was loaned out and used to purchase stuff). But with the loose definition you have $130M ($30M is still loaned out, depositors are $70M).
Essentially the depositors have made $100M of (debt) investments in the bank, and those investments are now 70% backed by cash and 30% backed by paper (mortgages or other kinds of IOU's from whoever they loaned the $30M to).
That's incorrect. A bank can't give out more dollars than they receive from depositors. It can give out all kinds of paper that is _valued_ in dollars, but not the digital representation of banknotes with the pictures of dead presidents on the front.
In contrast, the Fed can create actual dollars. It can just buy an asset and pay for it with money that it has just created.
Every time a bank makes a loan, the dollars are "created out of thin air", and slowly "destroyed" as the loan is repaid.