Neither Tigraine nor Taringail renounced their House when marrying. Once you understand this, things will be clearer.

Renouncing a House to integrate another is something done in Andor under Andoran law. It's not some universal law. It's not done in Saldaea, for instance. It's not done in Carhien either.

Morgase OTOH forced Taringail to marry into House Trakand and to renounce his Cairhienin House (under Andoran law). That didn't make him Trakand by name (it was strictly a matter of choice for him to use the name or not, I would guess. Elayne alluded once she could use the name Damodred the same way if she wished), but he was now an Andoran noble subjected to the High Seat of House Trakand under Andoran law (which clarified his legal position in Andor, and was an attempt to remove his standing within House Damodred in Cairhien. That was meant to but didn't work to stop Laman from using Taringail to plot in Andor. Obviously, Andoran law considered Taringail legally a member of House Trakand subject to Morgase authority as both HS and Queen, while with his marriage to Tigraine he had remained legally a Cairhienin, but this had no legal bearing in Cairhien and with Laman, for whom Taringail was still considered a member of House Damodred). It's never mentionned in the same breath that Morgase forced Taringail to bring his son by Tigraine into House Trakand too, she just formely adopted him, which isn't necessarily the same thing. As there's no real political reason why Morgase would have harshly forced Taringail to agree to the end of his line through Galad - unlike his father he was Andoran by birth already (while there were reasons to marry him into Trakand and have all his offspring from her to be legally born in Trakand), it's fairly unlikely Morgase did that (or that Taringail would have agreed). In any case, officially part of House Trakand by his adoption or not, Galad is allowed by law to keep using his Damodred name. Obviously, those Andorans laws about weddings, belonging to Houses or not, offsprings etc. have no legal bearing whatsoever in Cairhien, as Elayne could get her herself recognized as a valid sion of House Damodred by its High Seat Caraline. No one ever mentionned that the law that excluded Taringail stood in her way, so it sounds like the Law that excluded the consort of a foreign ruler from making a claim has overlooked the specific possibility a foreign ruler of Cairhienin descent coud make a claim too, likely because it was so preposterous no one foresaw a situation in which any Cairhienin House would ever support any such claim by a foreign ruler).

Mordrellen never married Tigraine "into Damodred", she simply didn't require from Taringail he married into House Mantear, and obviously she agreed that his eldest son would retain his Cairhienin name and Cairhienin house, even in Andor. That all most likely had to do with the complex negotiations for this wedding between Mordrellen and Laman (with no doubt the WT behind all this). By this wedding, Andor hoped to put an end to a series of recurrent wars between the two nations. Laman however had far more insidious motives to agree to it and hidden ambitions for Cairhien to grab control of Andor through this wedding. He meant for Taringail to be facto be co-ruler of Andor, and eventually ruler of Andor in his wife's place in all but name (Morgase would have had to change the laws of Andor for Taringail to eventually co-rule or rule officially). Elaida and Thom stood in the way of Taringail. They kept Taringail from gaining anywhere close to that kind of influence over Morgase. Taringail got officially appointed First Prince of the Sword, but as far as we know that appointment didn't come with the usual leadership over the Queen's Guard in time of War (it remained with Lord Naldwinn - who was likely Commander under First Prince Bryne in Mordrellen's time, and who died the same week as Laman, and Morgase appointed Mordrellen's former First Prince, Bryne, as Commander of the Guard, under Taringail's nominal/symbolic authority. After Taringail's death, Bryne was appointed again officially First Prince of the Sword, retaining his function as Commander of the Queen's Guard beside).