Per the FIDO spec they implement the protocol on top of the standard libUSB that's available as part of all current generation OSes. So yes, in a way, the OS does get involved, but only in so far as it treats it as yet another generic USB device. It's up to the application that implements the FIDO spec to send the proper commands over USB to interface with the HID device.
I do understand what they're saying, though. My bank also uses 2FA. You can type a challenge into a dongle, but you can also hook up the dongle. However, it needs a USB driver to work. Every OS update, it's uncertain whether it still works. I don't use it anymore because of that reason.
This stuff built into the browser makes it easier.