You make a good point regarding “how would the priest know who the person was”, but since Vatican II penitents can request absolution face-to-face or not. So, a priest can know who the penitent was.
Up to this point, I hadn’t thought of this specific example and only adds to my beliefs that changing to face-to-face confessions was a mistake.
I think it was more one of those damned "spirit" of V2 things. Confessions could always be done face-to-face, e.g. at a hospital, a jail cell, a convalescent home, in a war zone, The "spirit" of Vat 2 suggested that sin be de-emphasized and replaced with a more affirming/therapeutic approach, and anything that suggested especial seriousness or gravity or the solemnity of God's justice (e.g. semi-darkened confessionals as durable architectural features of a church) eliminated.
In most parishes' experience, the practice of Confession has dropped precipitously.
I don't know of this was ever mandated by actual Conciliar document.