“Only one “Fairy Archer” can have its effect activated per turn” is not treated as an effect.[1]
Even if multiple Fairy Archers exist, you can only activate the effect of 1 of them.[1]
Even if you change the name of Fairy Archer with a card like Hero Mask you cannot activate its effect more than once per turn.[1]
OCG Rulings
The text "The effect of "Fairy Archer" can only be used once per turn." of "Fairy Archer" is not treated as the effect of an Effect Monster.[2]
If you have multiple copies of "Fairy Archer" on your field, then you can only activate one copy's damage-inflicting effect, once per turn.[2]
Even if you change the name of "Fairy Archer" by an effect like that of "Hero Mask", then you still can only activate its damage-inflicting effect once per turn.[2][3]