This implies that the one who grants absolution must also be able to withhold it, based on the disposition of the penitent. That is why confession must be oracular - all serious sins must be told to the priest to the best of the penitent's ability to remember. True conversion necessitates that we recognize our sins for what they are and humble ourselves by revealing them in confession. The possibility of anonymous confession ought to relieve much anxiety, but confession always remains what the Holy Father more commonly calls it - Penance. By the way, it is very rare for a confessor to withhold absolution, so this possibility should not be a cause for fear.
Absolution is given to all sincere penitents.