When Pastors (whether Presbyters or Bishops) pronounce absolution whether following a general confession of an entire congregation, such as in the Anglican divine office or Holy Communion service, or an individual auricular confession as practiced by Anglicans, Lutherans, Orthodox, Catholics and many other denominations, they do so in the name of our Lord, God and Savior Jesus Christ, because it is Christ who is actually forgiving the sins. Indeed when engaging in auricular confession in Eastern Orthodoxy, one faces an icon of Christ, and venerates the cross after the Absolution.
What the Pastors are doing is using the “Office of the Keys” as some Calvinists call it, that is to say, the ability to bind or remit sins, to remit them, as per Matthew chapter 16, etc. But the people confess to Christ and are forgiven by Christ, for it is through His redemptive Passion on the Holy Cross their absolution and renewal is possible. And not only that, but also their sanctification, for Christ restored and glorified our human nature, and sent his Holy Spirit to be our Paraclete and Comforter, to dwell within us and enable us to attain Theosis. This is the faith clearly expressed by the early church, which is still taught by the Orthodox, and which was also taught by John Wesley, and also apparently Martin Luther expressed views favorable to it (and I recall
@hedrick stating that John Calvin expressed some views akin to Theosis). One might also argue that the Roman doctrine of the Beatific Vision is something akin to Theosis, although I am not at all sure about that, as much as I love the Catholic Church, I feel the Beatific Vision is an innovation of Scholastic Theology and a departure from Patristic theology, if I am understanding it correctly, and is underwhelming compared to Theosis, or Entire Sanctification as John Wesley called it.