Similar expressions are found in Scripture:5Slaves, be obedient to those who are your earthly masters, with fear and trembling, in singleness of heart, as to Christ; (Ephesians 6:5)
22Wives, be subject to your husbands, as to the Lord. (Ephesians 5:22)
14and though my condition was a trial to you, you did not scorn or despise me, but received me as an angel of God, as Christ Jesus. (Galatians 4:14)
Christian slaves were to be obedient and submissive to their masters as to the Lord. So, to Christian slaves, their masters were Christ on earth. Christian children are to be obedient and submissive to their parents, as to the Lord. So, to Christian children, their parents are Christ on earth. Christian wives are to be obedient and submission to their husbands, as to the Lord. So, to Christian wives, their husbands are Christ on earth. Christian lay people are to be obedient and submissive to the ordained ministers of the Church, as to the Lord. So, to Christian lay people, their ordained ministers are Christ on earth. Since the pope is the chief ordained minister of the Church, to all Christians, the pope is Christ on earth.