Is there any other authentication method like google authenticator for 2fa to use facebook?
[1] https://m.facebook.com/notes/facebook-security/security-key-...
You cannot enable U2F without first enabling SMS Auth and or OTP Auth. Once u2f enabled if you delete both SMS Auth and OTP auth it disables 2 factor auth and takes you to a welcome screen to get started enabling it again with the only 2 options SMS / OTP, once enabling one of them you can add U2F again.
Also I have "security.webauth.u2f" enabled in Firefox but Facebook is the only U2F service I use that doesn't work in Firefox so it falls back to the SMS / OTP methods on login.
I noticed this as well. I was able to add my security key via Firefox, but for some reason Facebook decided that you need Chrome to use U2F when logging in.