A more direct translation would be along the lines of “Phoenix’s Tail”, but even that is misleading because it’s referring to the tail feathers. The Japanese version didn’t specify the individual tail feathers, the same way English wouldn’t designate the individual hairs on a cat’s tail. A more accurate localization may be something like “Phoenix Flight Feather”, but that’s a bit of a mouthful (and would take up a lot of room in your game menus).