It's an exchange, the cryptocurrency itself is the bank. Banks don't require dollars or any fiat currency to exist, hell they invented paper money. The major difference is that banks had the advantage of gold and other precious metals as a pre-existing common medium of exchange. Cryptocurrencies need a verifiable way to say that you transferred X value from chain A to chain B. I'm sure this is a solvable problem.

> a mechanism that allows two completely unrelated & unconnected blockchains to transfer tokens between eachother

Atomic swaps already exist.

I still have no idea what that has to do with the nonsensical metaphor "a blockchain is a bank"

The essence of both a bank and a cryptocurrency is the ledger. Everything else is an implementation detail.

The essence of banks is dealing in risk. A ledger is one tool they use.