Can the money multiplier happen within a single bank / a single loan? Many tutorials explain the money multiplier effect by involving a chain of banks (e.g. khan academy)
So if reserve requirements are 10% and bank A owns 1000€ reserves, bank A can lend 900€ to bank B; now bank B has 900€ reserves so it can lend 810€ to bank C, and so on to infinity. Making it $M1 = \frac {1} {10\%} \times M0 = $ 1000€ reserves + 9000€ loans.
Is this chain really necessary, or can simply bank A with 1000€ reserves issue a single loan of 9000€?
From what I saw, the video confused various forms of monetary instruments.
If a customer gets a \$1000 transfer from the Federal Reserve, Bank A has a \$1000 deposit liability, and a new \$1000 balance at the Federal Reserve.
Under the assumption that the bank was at its reserve limit, it has \$900 in excess reserves.
It can eliminate those excess reserves by making a \$9000 loan, as that creates a deposit with \$900 of required reserves. The growth chain ends there.
In practice, the proceeds of a loan are typically spent quickly. As such, the bank would expect a need for an outflow of the size of the loan. It needs to hold liquid assets that can be liquidated to meet the outflow, or have the capacity to borrow in the inter-bank market.
Given the unpredictability of cash flows, in the pre-2008 era in the US (one of the few places with formal reserve requirements), banks made loans, and then borrowed in the Fed Funds markets to make up deficiencies. The Fed supplied the Fed Funds market the amount required to hit its policy interest rate target. (Currently, reserve requirements have been abolished, and there is a permanent excess of reserves.)
The presentater probably relied on out of date textbooks.  This paper by BoE staff explains how the “money multiplier” is a misconception: link to working paper.