This seems like your taking it a bit to the extreme. We both agree that for me to hire you to do a job I’m allowed to know who you are right? And that to identify the one person out of the world population I am allowed to keep a record with enough details that I can identify if a person is in fact you, right? Now if my bookeping of people changes from “roughly visually resembles the same person” which we both agree is a rubbish measure if any kind of security is involved to “has known fingerprint” and you refuse to comply, why then should I not be allowed to fire the person who I can no-longer identify out of the world population?
You have complete freedom to not want people to see your face or know your fingerprint, but you just cannot take away the freedom of a potential employer to be able to identify who you are by the means they decide.