In consumer devices the face data is stored only on-device, so any privacy concerns are defended by "it's not sending your face to the cloud" and/or "don't use it if you don't trust it"
At least with fingerprints you have 9 extra passwords you can use if your index finger print leaks (I bet my prints are already out there just from getting pre employment background checks from workplaces with hulking old school IT bureaucracies and a workforce pounded daily by phishing attempts).
I think it's the linkage of the name and the face. Either of those things alone don't seem like private information, but together, I would argue they are private. I can't provide a good reason as to why I think that yet though.
All I'm wondering is how this would play out in a dystopian future where I'm recognised by everyone wherever I go.
And in some cases they might even have additional information maybe in a probabilistic fashion. Like what kind of clothes I prefer or the kinds of food I like. And that would be the good version.
The worse scenario would be that some malicious actor puts all this together to play on others' fears.
The worst case, of course, being big brother!
So Faceapp knows "There's a person with this name who looks like this". Was that supposed to be a secret?
I'm concerned about privacy too, but it seems like some people just want to hide under a rock for their entire lives. I don't know how anyone manages to leave the house if having people see your face and know your name is a serious concern.