Opus Dei is a spiritual family, and the prelate is the head of the family. It wouldn’t do to have children call their father by some formal name such as “Msgr. Escrivá,” or today, “Bishop Eccheverría.” So when Opus Dei was beginning, the members would refer to Msgr. Escrivá as “the Father,” which sounds much more natural in Spanish (“el Padre”) than it does in English. The tradition has continued.
When St. Josemaría died Msgr. Álvaro del Portillo, also known as Don Álvaro, was elected to be the new head of Opus Dei. He became the new “Father” of the family, so we started calling him “the Father.” Of course, that would lead to confusion… Who would we be referring to when we used the term “the Father?” Would it be Msgr. Escrivá or Don Álvaro? So it was decided that Don Álvaro, or whoever was the head of Opus Dei, would be called “the Father” and the founder, Msgr. Escrivá, would be called “our Father,” in much the same way as the Franciscans, for example, might refer to St. Francis as “our father, Francis.”