As you will have seen in lots of other posts to this topic, people want privacy and "I just show my membership ID everywhere, what's the problem?" unsurprisingly is not what they had in mind.
So, FIDO preserves privacy by minting unique credentials for each site where you use it. This is invisible to the user of course, for them it's just the case that you use your FIDO authenticator everywhere (that it works) and now it's secure.