U2F is the predecessor to the current standard, WebAuthn. If a web app supports WebAuthn, then that integrates with native keystores (Windows Hello, Face ID, whatever Android has), as well as hardware keys. The site operator has some flexibility to prefer certain methods (platform vs. external) and devices (attestation).

I wrote an authenticating proxy that uses Webauthn: https://github.com/jrockway/jsso2. I don't think you should use it, but you can fire it up locally and try enrolling the various devices. Actually, you can just use Duo's demo: https://webauthn.io/