It does not. The fingerprint always only unlocks the device’s HSM (“secure enclave” in Apple speak).
Between your devices enrolled in the ecosystem, private keys are synced securely (AFAIK, they make it so that an existing device’s HSM encrypts keys using the pubkey of the new one’s HSM); for signing up using your device on someone else’s computer there’s a process that combines QR codes with Bluetooth communication.
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Note that you pretty much can’t store them with Google or Apple; smartphone biometric sensors operate the on-device HSM, not something remote.
So, how does it work when you are accessing account from a different device? How the other device knows your fingerprint?
It does not. The fingerprint always only unlocks the device’s HSM (“secure enclave” in Apple speak).
Between your devices enrolled in the ecosystem, private keys are synced securely (AFAIK, they make it so that an existing device’s HSM encrypts keys using the pubkey of the new one’s HSM); for signing up using your device on someone else’s computer there’s a process that combines QR codes with Bluetooth communication.