Could we ( Catholics ), say that " All the believers are Vicars of Christ " somehow. And not only The Pope and the Bishops ?
Would that be a heresy if someone believed such thing ?
In the Catholic Church, the "Vicar of Christ" is a role that is unique to the Bishop of Rome. Bishops, including the Bishop Rome, are seen as the successors of the Apostles, but only the Bishop of Rome is seen as the unique successor of St. Peter.
It would be heresy to say that anyone, even another Bishop, could fulfill this role. Only the Bishop of Rome as part of his office can inherit and take part in "the Petrine Ministry" which includes the title "Vicar of Christ". It would also be worth noting that St. Leo the Great referred to himself as a Vicar, but not of Christ. St. Leo saw himself as the "Vicar of St. Peter".
894 "The bishops, as vicars and legates of Christ, govern the particular Churches assigned to them by their counsels, exhortations, and example, but over and above that also by the authority and sacred power" which indeed they ought to exercise so as to edify, in the spirit of service which is that of their Master.426
1560 As Christ's vicar, each bishop has the pastoral care of the particular Church entrusted to him, but at the same time he bears collegially with all his brothers in the episcopacy the solicitude for all the Churches: "Though each bishop is the lawful pastor only of the portion of the flock entrusted to his care, as a legitimate successor of the apostles he is, by divine institution and precept, responsible with the other bishops for the apostolic mission of the Church."41