No, not necessarily. As seen with Cersei Lannister, a woman doesn’t necessarily take her husband’s name.
Also, based on some questions I got, a clarification/extension:
- Dornish custom is that Arianne’s kids take her name, so her family name is not in danger.
- Sansa is a bit more iffy, but there is precedent for husbands and children changing their names to maintain continuity: Joffrey Lydden became Joffrey Lannister when the male line ended early in the Andal period; the Tallharts offered to have Beren Tallhart (whose mother is a Hornwood) change his name to Hornwood if he was named heir, sot hat the line could continue.
I’ve always thought that Dorne’s equal-opportunity inheritance laws should lead to a lot of weird politics surrounding House names, words, and sigils.
Logically speaking, Dorne ought to experience a slow but gradual aristocratic consolidation, as in every generation 50% of heirs will be women (possibly a bit more, as women don’t go to war so much even in Dorne and therefore will die less) and some of those heirs will marry male heirs, uniting their houses and fiefs together in the person of their heir.
It makes sense from pure power politics standpoint (”Listen, the two of us together are getting pushed around, but if we marry our kids and unite our demesnes, our grandkids will be much more powerful”) but there’d be a lot of haggling over which names and symbols are passed on.
Or there might be a social taboo against heirs marrying heirs to prevent consolidation/loss of fiefdoms, to compensate.