2 Edited to mention TOTP is a good fallback method and should not be discouraged.
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Short answer: Some sites allow you to turn off your backup option if using a U2F device, but doing so is at your own risk.

Long answer: Convenience and security are often at odds with each other. You are correct, having a backup login solution defeats the purpose of having the U2F key, as technically you could use your fallback method to bypass it. Any account is only as secure as its fallback authentication. If there is no fallback, then you are secure. You are also without recourse if you lose your U2F device. To my knowledge, no two U2F devices are created with the same private key, or at least they shouldn't be. If you had two duplicate U2F devices, this too would be a weakness in your security. It should be noted that a TOTP application is a relatively secure fallback as long as the device you're using is secured properly.

Short answer: Some sites allow you to turn off your backup option if using a U2F device, but doing so is at your own risk.

Long answer: Convenience and security are often at odds with each other. You are correct, having a backup login solution defeats the purpose of having the U2F key, as technically you could use your fallback method to bypass it. Any account is only as secure as its fallback authentication. If there is no fallback, then you are secure. You are also without recourse if you lose your U2F device. To my knowledge, no two U2F devices are created with the same private key, or at least they shouldn't be. If you had two duplicate U2F devices, this too would be a weakness in your security.

Short answer: Some sites allow you to turn off your backup option if using a U2F device, but doing so is at your own risk.

Long answer: Convenience and security are often at odds with each other. You are correct, having a backup login solution defeats the purpose of having the U2F key, as technically you could use your fallback method to bypass it. Any account is only as secure as its fallback authentication. If there is no fallback, then you are secure. You are also without recourse if you lose your U2F device. To my knowledge, no two U2F devices are created with the same private key, or at least they shouldn't be. If you had two duplicate U2F devices, this too would be a weakness in your security. It should be noted that a TOTP application is a relatively secure fallback as long as the device you're using is secured properly.

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source | link

Short answer: Some sites allow you to turn off your backup option if using a U2F device, but doing so is at your own risk.

Long answer: Convenience and security are often at odds with each other. You are correct, having a backup login solution defeats the purpose of having the U2F key, as technically you could use your fallback method to bypass it. Any account is only as secure as its fallback authentication. If there is no fallback, then you are secure. You are also without recourse if you lose your U2F device. To my knowledge, no two U2F devices are created with the same private key, or at least they shouldn't be. If you had two duplicate U2F devices, this too would be a weakness in your security.