If you recall back in TDR, when Rand is facing Be'lal, there is this passage:

The Forsaken laughed, amused, swinging his blade in quick flourishes to either side of him; the black fire roared as if swift passage through the air quickened it. “You were a greater swordsman, once, Lews Therin,” he said mockingly. “Do you remember when we took that tame sport called swords and learned to kill with it, as the old volumes said men once had? Do you remember even one of those desperate battles, even one of our dire defeats? Of course not. You remember nothing, do you? This time you have not learned enough. This time, Lews Therin, I will kill you.” Be’lal’s mockery deepened. “Perhaps if you take Callandor, you might extend your life a little longer. A little longer.”

So, when Rand said the sword felt familiar to him, I assumed it was LTT's, but that's just me. However, it would offer some explanation for it being a Power-wrought sword minus the Heron-mark, which perhaps came into use sometime after the "sport" became popular.