Strictly speaking, it is not correct for anyone to introduce themselves by title. Traditionally, it is considered a mark of modesty not to refer to yourself by title. That is for others to do. And, in any case, the congregants don’t really need to know the personal identity of the presider; they usually already know that anyway. A priest presides at Mass in persona Christi, so the primary presider is Christ himself. It is considered helpful for visitors to know the name of the presider though, so that is usually done when one of the lectors says before Mass something to the effect of, “Our celebrant today is Father Patterson; he is assisted by Deacon Lindstrom.”
As for a priest introducing himself to others outside of Mass, it is not totally wrong for him to use his title “Father,” but he could just as easily say, “I’m James Patterson, pastor of St. Bartholomew Catholic Church,” and trust those he meets to supply the “Father.” If someone instead takes this as permission to call the priest “James,” he could say, “Please call me Fr. Patterson, Mr. Riley.” (Of course, if Fr. Patterson is a member of a religious order, the custom is to call a religious order priest by a title and first name; i.e., “Fr. James.”)
