Being a bit boring but....
If you pay the wrong amount of money to another account, or pay the wrong account, the person/company that got the extra money has to give it back. What the banks say is that they won't get it back for you. Which, if you think about it, is actually reasonable. The banks most certainly have the capability to reverse a transfer (although if it went between banks it is messy) but the last thing you want is the bank allowing people to reverse transfers on request.
It is the same mess that cancelling cheques gets you into. It isn't the banks job to go after your mistakes, but far more importantly, it isn't the banks role to provide an easy mechanism to allow less scrupulous people to reverse any transfers they make after the fact. This would eventually force the bank to act as arbiters in disputes. That isn't their job.
Of course if the money that moves by mistake is the bank's own, they will move it back. But here they are one of the parties of the transaction, not the mediator.
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