I don't think there is an ASME spec for the packing. The pervasive practice in the industry is to call out the spec number from the packing manufacturer used, unless the valve manufacturer just wants to keep the packing spec proprietary.
Example: Masoneilan (I do not work for them) identifies their packing as Crane 285K. This particular packing is a square-cross-section aramid fiber (Kevlar) skive-cut ring with a TFE overbraid. For an all-braided TFE ring, they call out Chesterton 324. Other packings are just called out as TFE V-ring or Grafoil.