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.
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.