|Maybe, maybe not.|
The phrase itself appears to be borrowed from St. Paul's second epistle to the Corinthians, wherein he writes, "I myself, wherever I have shewn indulgence, have done so in the person of Christ for your sakes" ("nam et ego quod donavi, si quid donavi, propter vos in persona Christi").
It appears in its more recognizable form in St. Thomas's Summa, where he distinguishes between Sacraments: some being effected in the person of the minister (e.g., "I baptize you..." or "I confirm you...") and others in the person of Christ (e.g., "I absolve you..." or "This is my Body..."). For Thomas, this appears to be a technical point concerning the meaning of the words effecting these Sacraments, rather than an observation on the effects of Holy Orders on ordained persons.
The Catechism of the Council of Trent (1566), though not quoting the council itself, says, "The priest is also one and the same, Christ the Lord; for the ministers who offer Sacrifice, consecrate the holy mysteries, not in their own person, but in that of Christ, as the words of consecration itself show, for the priest does not say: This is the body of Christ, but, This is my body; and thus, acting in the Person of Christ the Lord, he changes the substance of the bread and wine into the true substance of His body and blood." In this, the catechism is basically following Thomas.
Whereas previously the bishops held a special place as the ministers and missionaries of Christ, the Counter-Reformation seems to have gradually extended of this idea to all priests. St. Alphonsus Liguori (†1787), for instance, wrote tracts on the dignity of the priesthood, at one point opining, "For us it is enough to know that Jesus Christ has said that we should treat his priests as we would his own person."
Similarly, St. John Vianney (†1859) wrote, "After God, the priest is everything! Only in heaven will he fully realize what he is." Like sentiments are found in popular moral manuals and spiritual writings from the Counter-Reformation to the second Vatican Council.
The twentieth century saw an increase of exposition on the exaltation of priests. Even James Joyce's Portrait of the Artist as a Young Man (1916) famously depicts a priest trying to encourage the eponymous hero to pursue Holy Orders based on its magnificent dignity: "No angel or archangel in heaven, no saint, not even the Blessed Virgin herself, has the power of a priest of God: the power of the keys, the power to bind and to loose from sin, the power of exorcism, the power to cast out from the creatures of God the evil spirits that have power over them; the power, the authority, to make the great God of Heaven come down upon the altar and take the form of bread and wine."
P. Pius XII wrote in the encyclical Mediator Dei that "Christ is present at the august sacrifice of the altar both in the person of His minister and above all under the eucharistic species" and that priests "represent the person of Jesus Christ before their people."
The Vatican Council increased the danger of clericalism even while giving lip service to the importance of the laity. In Lumen Gentium, the council fathers wrote that the bishops "in an eminent and visible way sustain the roles of Christ Himself as Teacher, Shepherd and High Priest, and that they act in His person." Also that the priests "by reason of their sacred office represent the person of Christ."
In the recent Catechism of the Catholic Church, the phrase is mentioned three times (875, 1348, & 1548), albeit modified into, "in persona Christi capitis." The extension of the persona Christi theology goes even further in 1348: "It is in representing [Christ] that the bishop or priest acting in the person of Christ the head (in persona Christi capitis) presides over the assembly, speaks after the readings, receives the offerings, and says the Eucharistic Prayer." This is the first "official" suggestion that a Sunday sermon should be considered as being preached in the person of Christ.
In 2009, P. Benedict XVI issued a motu proprio in which he modified the Code of Canon Law to clarify that only bishops and priests act in the person of Christ, not deacons.
While I am no expert in the development of doctrine, it seems that the relatively modest proposal of Thomas Aquinas got quite out of hand by the 20th century. His theological writings on the words of consecration have also caused no end of consternation with the various eastern Churches. It seems clear at this point in time that they contributed very unfortunately to the culture of clericalism, the quasi-quietism that idolized obedience even for the common layman.
One might say that a bishop acts in the person of Christ much like an ambassador acts for his country: in an official capacity, but sometimes poorly representing the desires of his fatherland, and being occasionally the cause of international incidents. Indeed, the aforementioned Pauline epistle calls the apostles "ambassadors of Christ," but it seems that St. Thomas had a very different meaning in mind when opining on the effecting of Sacraments.
P. Benedict's legal clarification is welcome, but does little to undo the damage of excessive clericalist spirituality. The recent sex abuse crisis saw many priests manipulating young men by asserting their role as "another Christ," and by claiming that whatever they willed was therefore willed by God. This is an extreme example of clericalism gone wrong, but not an irrelevant one. When your local parish pastor starts asserting his authority as an alter Christus, even for silly things like liturgical preferences in the Pauline Novelty Mass, but especially for more serious matters like demanding complete obedience to all advice given in the confessional, one must wonder why.