The security of the browser implementation is important. It provides the origin for the security hardware to sign, and the authenticating server ("relying party") verifies it. If your browser tells the key it's google.com when it's really evil.com, then sure, you can log into google.com if the user signs the request.
The WebAuthn spec says: "Direct communication between client and authenticator means the client can enforce the scope restrictions for credentials. By contrast, if the communication between client and authenticator is mediated by some third party, then the client has to trust the third party to enforce the scope restrictions and control access to the authenticator. Failure to do either could result in a malicious Relying Party receiving authentication assertions valid for other Relying Parties, or in a malicious user gaining access to authentication assertions for other users."
If you click further into the older FIDO spec, they cover this more explicitly: "Malicious software on the FIDO user device is able to read, tamper with, or spoof the endpoint of inter-process communication channels between the FIDO Client and browser or Relying Party application.
Consequences: Adversary is able to subvert [SA-2].
Mitigations: On platforms where [SA-2] is not strong the security of the system may depend on preventing malicious applications from being loaded onto the FIDO user device. Such protections, e.g. app store policing, are outside the scope of FIDO."
The WebAuthn spec says: "Direct communication between client and authenticator means the client can enforce the scope restrictions for credentials. By contrast, if the communication between client and authenticator is mediated by some third party, then the client has to trust the third party to enforce the scope restrictions and control access to the authenticator. Failure to do either could result in a malicious Relying Party receiving authentication assertions valid for other Relying Parties, or in a malicious user gaining access to authentication assertions for other users."
(https://w3c.github.io/webauthn/#sctn-client-authenticator-pr...)
If you click further into the older FIDO spec, they cover this more explicitly: "Malicious software on the FIDO user device is able to read, tamper with, or spoof the endpoint of inter-process communication channels between the FIDO Client and browser or Relying Party application. Consequences: Adversary is able to subvert [SA-2].
Mitigations: On platforms where [SA-2] is not strong the security of the system may depend on preventing malicious applications from being loaded onto the FIDO user device. Such protections, e.g. app store policing, are outside the scope of FIDO."
(https://fidoalliance.org/specs/fido-v2.0-id-20180227/fido-se...)
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