How does a bank lend your money while you have constant access to it?

Fractional Reserve Banking, that's how, under the assumption that only a small percentage of customers will take all -- or even most -- of their money out at any one time.

The country's central bank will mandate that banks maintain a fraction (typically 10%) of their assets in reserve for situations where a lot of people all at once want cash.

That used to happen quite often in the US, where everyone would run to the bank to get their cash. These "bank runs" would destroy banks, leaving other depositors with nothing. The FDIC was formed in 1933 to assure people that their money would always be there; since then there have been no (or very few, and limited in scope) bank runs in the US.

(In March 2020, the Fed reduced the reserve to 0% to help stimulate the economy.)

Banks are required to keep a certain amount of money in reserve to handle the expected rate of withdrawals. Banks know that while anyone can withdraw their money at any time, not everyone will withdraw their money on any day. So banks only need to keep a small percentage of the amounts deposited and can lend out the rest. If more people than expected want to withdraw their money, banks can also borrow funds from other banks and the Fed. If there was ever a situation where lots of people simultaneously wanted to withdraw lots of their money from lots of different banks all at the same time, the Fed would make sure that the banks remained sufficiently liquid to satisfy the requests.

Prior to the Great Depression, bank deposits weren't insured so if the bank went under depositors lost their money. That meant that if there was even a rumor that a bank was having financial trouble, people would literally run to the bank to withdraw all their funds. Everyone trying to withdraw their funds at the same time would almost inevitably cause the bank to fail even if it was perfectly healthy. Today, if you're putting money in a savings account at a bank, that money is going to be insured by the government (the FDIC in the US) up to a pretty ridiculous limit ($250,000 per depositor per bank per account ownership category). If your bank goes under, either the FDIC will step in and give you your money or, much more likely, your account will get sold to a different bank where you can withdraw your money. Deposit insurance has made bank runs much, much less likely.

If you're putting your money into something that is not a bank (or not a simple bank account at a bank), you'd need to look to see whether that account is insured. If a US bank lets you open an investment account, that investment account would be insured by the SIPC. That would protect you against the bank failing but not against the value of the assets in your account dropping. If you put your money in a money market account at a bank, the FDIC ensures you won't lose money. If you put your money in a money market fund at a brokerage account at that same bank, the SIPC protects you in case your bank fails but not in the unlikely event that the assets the money market fund invests in decline in value and the fund itself loses money.

If you're looking at a random fintech company, you'd need to look to see whether and how that particular company's accounts are insured to understand what risks you are taking with your savings in the event of a run.

Because there's no reason everyone would withdraw all of their money at once.* Thus you can model inflows and outflows from people as approximately independent random variables, statistically model them, and be able to make similar predictions that insurance companies make. After all, insurance companies also don't have the money to cover extremely rare events. If an auto insurance company has 100,000 customers and they all wrecked their car tomorrow, the insurance company wouldn't have the money to cover all of them. Of course, the likelihood of such an event is so small that it's essentially impossible. Similarly, everyone taking their money out of the bank tomorrow is extremely unlikely.

The problem comes when units that are modeled as statistically independent no longer act as such. A poignant example of this was the housing crisis that began in 2007/8. Normally, you can model the foreclosures of houses as independent random variables. However, this idea broke down once the housing bubble burst and prices plummeted. Houses went into foreclosure which lowers the prices of similar houses and makes them more likely to go into foreclosure. Speculators who were betting on constantly rising prices failed on their payments and everything combined leads to a vicious cycle that drives prices far below where they should be and everything comes crashing down. Needless to say, the banks did not even have a fraction of the money required to account for such a crash.

*A similar thing was true for banking prior to the reforms instituted after the start of the great depression. Before then, if a bank went broke everyone who still had deposits in it lost everything. This creates what is known as a self fulfilling prophecy. Even if a bank was actually financially healthy, if people thought a bank was going fail they would rush to get their money out. This caused other people to rush to get their money out and was known as a run on the banks, which was essentially a death sentence for even healthy banks. This lead to the dreaded scenario given in the OP -- a bank would be unable to cover their debts because everyone rushed to take their money out at the same time. But the FDIC was created and the risk on the customers side vanished. Thus, no more worry about mass withdrawals.

According to economist A. Werner and his case study this is because the banks do not need your money to create new debts..Excerpt:

An empirical test is conducted, whereby money is borrowed from a cooperating bank, while its internal records are being monitored, to establish whether in the process of making the loan available to the borrower, the bank transfers these funds from other accounts within or outside the bank, or whether they are newly created. This study establishes for the first time empirically that banks individually create money out of nothing. The money supply is created as ‘fairy dust’ produced by the banks individually, "out of thin air".

You think A. Werner is some sort of crank? Try reading this paper by Kumhof and Jakab published by the IMF (International Monetary Fund):

Specifically, virtually all recent mainstream neoclassical economic research is based on the highly misleading “intermediation of loanable funds” description of banking, which dates to the 1950s and 1960s and back to the 19th century.We argue instead for the “financing through money creation” description, which is consistent with the 1930s view of economists associated with the Chicago School. These two views have radically different implications for a country’s macroeconomic response to financial and other shocks. This in turn has obvious relevance for key policy choices today.­

Lending, in this narrative, starts with banks collecting deposits of previously saved real resources (perishable consumer goods, consumer durables, machines and equipment, etc.) from savers and ends with the lending of those same real resources to borrowers. But such institutions simply do not exist in the real world.

EDIT: While it is true that banks do indeed need some of the money they lend it is only a fraction of the debts they create. These are the reserve requirements. Countries like Sweden, New Zealand or Australia have ZERO percent reserve requirements. In the Eurozone it is 1%. In the US depend on the amount of debt and range between 3% and 10%. (see wiki article provided)

You don’t actually have constant access to your money, it just seems like you do. You have constant access to a pool of money that’s set aside by the bank to cover the likely demand for withdrawals, but is very much smaller than the total amount deposited with the bank. In unusual circumstances, that pool of money can run out, and the bank goes bust. Government guarantees will protect most small savers.

Banks don't precisely loan you somebody else's money. Rather, they pool its liquidity.

'Money' in general economic terms is any credible and enforcible promise to pay something of real value (e.g. goods and services) at a later date. It includes IOUs, vouchers, share certificates, contracts, etc. as well as cash. When you go to a bank and take out a loan, you sign a piece of paper promising to repay the loan on a certain schedule - signing it turns the loan agreement into 'money', which the bank puts in its vault. That is to say, it is the borrower that creates the money; that makes the promise. The bank then gives you an equivalent amount of money (minus their fee, discount for the risk of default, etc.) in return. They are in effect paying you back your own money that you have just created and given them. The value of the loan agreement sitting in their vault covers the loan 100%.

The real difference between the money you give them and the money they give back is a property called 'liquidity', which is roughly speaking all about how quickly you can turn it into something of real value. The loan agreement is a contract with genuine financial value, but it can only be realised gradually over the next 20 years, or whatever the period of the loan is. You want money you can spend now. This is where the savers come in.

Where a borrower exchanges an illiquid form of money for a liquid one, a saver does the reverse - giving the bank money in a liquid form and getting back a less liquid one: a contract with the bank promising to pay it back on demand. Liquid money can always be spent immediately, but usually isn't. People only need a bit of it at a time. So by pooling all the savings of lots of people together, the liquidity only has to cover the small fraction in motion at any given time, allowing some margin for random variation. The rest of the liquidity is available to be spread over the illiquid money deposited by the borrowers.

If all the savers demanded their money back at once, in principle the bank always has assets with value to cover the full amount. But because those loan agreement don't pay off immediately, their only recourse is to sell the illiquid debt to another bank in exchange for liquid cash. And there is a limited market for that - dump too much on to the market at once, the price will crash, and the troubled bank will only be able to get a fraction of the face value on those loan agreements. That's what often causes a run on the bank to lead to a crash. By promising to buy debt at close to face value, the central bank props up the banks and prevents crashes. (It can also happen that a bank invests customers' money acting as an agent for the customer, taking on the risk of investment itself, the investments drop in value, and the bank can become genuinely bankrupt. That's a different problem, and not necessary to its function as a bank.)

Banks don't create money, ordinary people do. It is created by those ordinary people promising to work hard every day for the next 20 years or so to pay back the loan - something of genuine and honest value. It is what backs the vast majority of the money in the economy. Banks trade liquidity, acting as intermediaries between savers with liquid cash they don't immediately need, and borrowers with immediate needs willing to make long-term promises. They don't transfer money/value from one to the other, only liquidity, which is why the monetary amounts don't have to be equal.

How do banks / fintechs lend your money while you have constant access to it, ie. can withdraw at any moment? I assume there is some kind of liquidation pool, but how exactly is that managed with limited risk?

So long as the bank is not insolvent and has significant assets that exceed its liability, a liquidity crunch is never a real problem. For example, say you demanded $10 and the bank didn't have it. They could simply offer you a coupon good for $11.50 in 30 days, with 18% APR interest to be paid if they weren't able to satisfy the coupon in 30 days. Since the bank is solvent and has significant assets that exceeds its liability, this coupon is worth at least $10, and you could simply cash it at the bank across the street who would be happy to have it.

The real problem is if the bank actually goes insolvent due to making lots of bad loans, not that it might have some short-term liquidity problem even while it has a valuable portfolio of loans and other assets.

It is commonly said that FDIC insurance prevented bank runs. That is true. But it's not because it protected banks from illiquidity. It's because the cause of a bank run is people afraid they'll lose their money because the bank is insolvent due to making bad loans. Illiquidity is only a minor inconvenience to a bank if it's solvent -- it just has to pay above market interest for a short period of time.

Focusing on the "constant access, can withdraw at any moment" is a red herring. Any solvent entity can solve liquidity problems by issuing debt -- who wouldn't want to lend money at well above market rates to a solvent bank with assets way beyond its obligations? The real problem is banks going broke because they made bad loans.