Wouldn't this not be prisoner's dilemma then? If I pull my money out, I'm in the same spot as before with my money parked at a different bank. If I don't pull my money out, I'm either in the same spot as before or even worse off.

In this case, starting from the perspective I have a choice (and thus the bank is still liquid), I can only do at best the same as I would have done if I pulled out.

Prisoner's dilemma requires that the parties cannot communicate with each other.

Communication is meaningless in prisoner's dilemma.

The famous example illustrating PD created by Albert W. Tucker does have no communication - so that parties don't coordinate and cooperate with each other - but in this case with thousands of parties, there was no way to coordinate them all, or communicate with them all in time, or trust them all, so it still counts. It's possible to set up prisoner's dilemmas with communication, say in cases where it's a one off and the parties have every reason not to trust each other's word.