You’re confusing the money itself with the institutions that manage the money.

I don't think so. They are creating the metaphor that banks == blockchain implementation + stakeholders, money == blockchain tokens. Currently most of the cryptocurrencies you see with ICOs are just erc20 tokens ontop of ETH (therefore centralized). For banks they leverage SWIFT + other more local mechanisms (double entry accounting, etc...) to resolve transfers.

Its possible in the future we will develop a mechanism that allows two completely unrelated & unconnected blockchains to transfer tokens between eachother. I assume there would have to be either an intermediate chain specific to this purpose or both chains would need to follow a similar implementation. So far we started with Proof Of Work, we now have Proof of Stake and Proof of History. There may be more mechanisms in the future.

Does your bank deal in dollars or its own token?

Is a cryptocurrency exchange more like a bank or a currency?

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.