He never became a Trakand. Marrying into another House doesn't change your name apparently, just decides what names your children will have.

She allowed him to become a Damodred, only to marry into Trakand? Read again Dobraine's quote. He feels Morghase should have married into Damodred, and that she needed Taringail enough that he could have forced her to do it.

What likely happened is this:
Laman was still interested in preserving the peace with Andor, and wanted Taringail married to the next Queen of Andor. We also know, from the guide, that Laman wanted his favored nephew to be co-ruler of Andor, and eventually sole ruler. That last was likely why Taringail married into Trakand. He saw that Morghase was weak, and he thought that by marrying into the Royal House, he could eventually gain enough support to kill her and assume the throne himself.

Here's the quote on Laman wanting Taringail to be co-ruler:



There's a good explanation for why he married into Trakand. He never married into Mantear because there was no chance of him taking over from Tigraine, whose political position was far more secure. Of course, Laman and Taringail plotted without accounting for one Thomadril Merillin and Morghase's own great leadership skills.


This is true for Galad, not Taringail. The Law of Cairhein prevented him from becoming King.