Perhaps you can apply the famous Ketzos that Hoda’a is also a Kinyan.
I do not want to go there. I would even rather say that the hishtamshus of the loan has a din of Kesef (similar to an old style gold/silver dollar bill?), and the transfer even within the system is a Maas’e Nesina (which is plausable), and the “Kinyan” is done similar to the gemorah Gittin 13B:
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I.E. part of the loan to the bank is that the bank can transfer the money to others at your discretion. (a much more “ba’al Habatish answer).