For FIDO (and thus WebAuthn, and thus to make this actually practical beyond a toy that only works for some particular Yubico product) the keys are
random per enrollment. This is intentional because it means that you can't be tracked, since "your" key on Facebook and "your" key on GitHub are no more related to each other than "my" key on Facebook is to "your" key on GitHub.
Google have apparently some plans to address this problem in the medium term. Adam Langley has written vaguely on this subject before. In the short term, their priority is the trick he wrote about most recently - if your Android phone is enrolled as a Security Key with Google, and it's signed in to Google because it's an Android phone, and you use Chrome on a desktop, which is also signed into Google, the Chrome can use Bluetooth to determine if the phone is physically nearby and if so propose to authenticate your desktop Chrome to a remote web site using the Android phone. Elegant, albeit not suitable for those who fear lock-in.
