It seems to be more complex. While spiritual fatherhood has always been part of the Church as referred in the bible,it has in the beginning been used for Bishops.

As for Priests, it seems to have been different from nation to nation, but the most common was lord(dominus), this can still be seen in romance languages with the title Don/Dom or monsignor(My lord). 

Calling a priest father seems to be of a more affectionate nature similar to pastor. In any case in the East there also exist a tradition of calling priests father batushka(батюшка), (oytets)отец, and even pop(поп) in several Slavic languages. Likewise in Greek.

Curiously in Arabic/Aramaic it is historically been different, where Bishops are refereed to as "master" but priests and hegumons as abouna/abba(father). Where the word Abbot comes from. 

There is also the Latin rite where the servers refer to the priest as "pater"
Suffice it to say I don't think I would call it a recent innovation.