Well, we can actually determine a fair bit of information about family naming from indirect sources, one of the best is the official naming traditions thread.

But to summarize, hyur clearly take family names (the Hildibrand quests are actually an example of this, although who actually took the name Manderville when Godbert and Julyan married is unclear). The naming conventions thread also made note of the padjali naming conventions referring to how certain Midlander families in Gridania tend to produce multiple padjal and thus when a Midlander child is chosen by the elementals to become a padjal they retain their Midlander family name (the Senna siblings are an example, 'Senna' is their Midlander family name).

As for elezen, again like the hyur they clearly adopt family house names to the point of zealousness, but Ishgardian elezen especially have their own traditions in this (not mentioned in that thread, but obvious during the Heavensward storyline). Specifically, in the Ishgardian nobility a family name is only taken by the children of the family if they are legitimate offspring. Illegitimate children however are sadly forbidden to take the family name, instead bearing a nondescript moniker (Haurchefant is an example, he was not given the Fortempts family name but instead the generic, somewhat Midlanderesque surname 'Greystone', although Edmont loved him all the same - Atoriel in particular stating Edmont had a soft spot for him and "never could say no to my dear brother".).

An exception to this is Aymeric, whose surname is de Borel (so presumably his family name), yet he too was revealed during HW's story as also being illegitimate (although Aymeric's status as a bastard child is unusual, as he was born to a member of Ishgard's clergy (who, like the modern Catholic church, are forbidden to marry or bare offspring), and thus the rumours of his parentage have hounded him all his life (and confirmed at the end of the Vault when Aymeric yells "FATHER!!" to Archbishop Thordan). Hence it's likely Aymeric was given his mother's name as a way of sweeping proof of Thordan's indiscretion under the rug.

Elezen outside of Ishgard (read: Gridania) seem to follow a similar naming pattern. Louisoix's grandchildren after all share his surname, and this suggests that traditionally elezen children at least seem to take their father's name, but there is no 100% proof of this.