The handle (IIRC the FIDO spec calls it a cookie but that doesn't matter) is NOT chosen by the relying party (in this case a SSH server you're logging into). It's chosen by the token itself during the enrollment process and given to the relying party with the other results of enrollment to be stored.
The token will use local random data in addition to random data supplied by the relying party for choosing a private key (and thus in the design you describe, the handle). This is a common design choice in cryptographic systems because it means if either Alice OR Bob really used random numbers the results are truly random, things only go badly if both parties defect.
In WebAuthn this means if you use your token to enroll with Google, and Facebook, and say Twitter, so long as the people who built the token did a good job it won't matter if both Google and Facebook are trying to betray you and deliberately didn't use random data, they can't attack your Twitter account this way.
Edit: refer to the device as a "token" not a key for clarity