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Atleast for my u2f token, I'm being shown the site I'm signing for on a hardware screen.

Which device are you using? With U2F, the browser doesn't send the name of the site to the authenticator.

I'm using Trezor, I believe it has been preloaded with certain websites so it knows Github and Google and the likes.

It also shows parts of the public key (or so I believe, it is a unique identifier) per website.