The title descends from the Breaking of the World and the formation of the White Tower. At the time, the Seven Seals were dispersed to prevent male Aes Sedai or Shadowspawn from finding them. However, sometime during the Trolloc Wars, knowledge of these hiding places had been lost, and the White Tower apparently never actually possessed them. Since then, while no Amyrlin has known the location of the seals, they still claim the title.
As Rand al'Thor later pointed out, the title is essentially symbolic and was practically meaningless, as the White Tower neither possessed the seals nor had any Amyrlin Seat ever really watched over them.
Anyway just prior to the Last Battle finally the Amyrlin Egwene fulfilled somehow the role of Watcher of the Seals when Rand was convinced to give her the three remaining intact Seals (even if later it was revealed that they were not the real ones because of a manipulation by M'Hael). During the Last Battle Egwene also explains just a moment before to die when it is the right time to break the three real Seals, permitting the Dragon to accomplish his mission.