Using a U2F USB authentication key to login to services such as Google seems a good idea. However, these services often allow you to register a backup two-factor-authentication (2FA) method, which you can use in the event that you lose your U2F physical device.
The fact that you can just use a 'backup method' to login to these services seems to defeat the point of having the U2F key in the first place. If security is only as strong as the weakest link, then a hacker with password access to an account can just bypass the U2F device by claiming that the key went missing, using (for example) SMS.
On the other hand, people may not want to risk all their web account access to a single USB device. If it's attached to their keys, there's every chance it could get lost, left out in the rain, etc etc. Ideally, it would be possible to purchase two U2F devices that are registered with the identical 'secret'. By not allowing any other 2FA method, then in the event that the U2F device went missing, I would have a backup securely stored at home.
Question: Is it possible to duplicate U2F devices and set online services (such as Google) to only allow U2F, with no other backup method? Obviously if a user were to buy two U2F devices and they were to both go missing, then the user would be irreversibly locked out of their online accounts, but this would be a risk that the user would knowingly sign up to. I really can't see the point in switching to U2F devices unless this is an option.