> What's stopping me from using a browser to log in to my bank?

The "security module" they require you to install on your computer. In the past, when browsers had plugins, this was a browser plugin; nowadays, it's an always-on service (running as root) which exposes a local HTTP server which the bank site connects to to validate your computer. For an example from a major bank in this country (the same "security module" is used by several banks in this country), https://seg.bb.com.br/home.html is the diagnostic page for that "security module" (the FAQ page there has links to the installers).