Banking doesn't strike me as a use-case that the government needs to worry about, since a mandate already exists to report financial information and suspicious banking activity to law enforcement. A backdoor would be more
convenient for them, but encryption of your communications between you and the bank does not mean the bank is encrypting your financial and demographic information in such a way that they themselves will not be able to retrieve it.
What's being discussed here is the Apple strategy, where the service provider can provide a service to the public without the ability to retrieve information generated by the service.
The desire for companies to provide such services was likely spurred by the government's insistence that they provide private encryption keys, even in the event that they provide access to encrypted information not limited to the direct subject of a warrant, and the federal courts' repeated confirmation of the government's power to enforce such requests. See the demise of Lavabit [1].
There are numerous technical workarounds to this problem; for example, a private-key infrastructure that generates one key pair per message, or per customer. However, none of those workarounds allow a service provider to market products that are end-to-end secure from eavesdropping or the later revelation of the customer's information to empowered third parties.
So in essence, the argument I'm making is that it doesn't matter if there is a fundamental difference between banking and a chat application, because in either case the government either already has or is actively seeking the power to obtain the information you generate.
As it's hard to imagine a world where malicious parties would not be able to obtain encrypted communication mechanisms (unless you ban open-source, computers, pens, and the human brain), it's difficult to reconcile the above with the proposed effect this law would have, and it's very difficult to imagine the government restricting the application of these new powers to terrorist organizations only. See the history of the Patriot Act, for example [2].
[1] https://en.wikipedia.org/wiki/Lavabit
[2] https://www.washingtonpost.com/news/the-watch/wp/2014/10/29/...