Sokka and Katara's relationship in Avatar: The Last Airbender is key to Aang's success, but there's one shared tragedy that is the basis of their strength. The two siblings are the first characters introduced in Nickelodeon's Avatar: The Last Airbender, with much of the action taking place in the Southern Water Tribe, where Sokka and Katara hail from. Beyond the divisions of the four nations in the world of The Last Airbender, the Water Tribes have more divisions between their own peoples, with most waterbenders hailing from either the Northern or Southern Water Tribes, while there's a small minority from the Foggy Swamp Tribe.
While the Water Tribes have some differences between them, they share most of the same customs and beliefs, such as their high ideals of family and hospitality. Nonetheless, there are some incidents of tension between the tribes, most noticeably the attempt of Avatar Korra's uncle Unalaq to seize the Southern Water Tribe in Legend of Korra. Despite this, Sokka and Katara's visit to the Northern Water Tribe in The Last Airbender generally displays the mutual respect of the two tribes, who consider themselves kin.
F.C. Yee's novel The Dawn of Yangchen, which chronicles the Air Nomad Avatar before Aang, hints at one Water Tribe custom that is mostly absent in The Last Airbender. Upon meeting each other, two waterbenders, named Kavik and Tayagum, introduce themselves in a specific manner: "Tayagum. Of the Orca Islands, after my cousin. My parents are Angtan and Taganak." Since Kavik identically reciprocates the introduction, it's safe to assume that this is the standard protocol between waterbenders, wherever they're born. There is no point in Avatar: The Last Airbender, however, when Katara or Sokka use such an introduction. The closest is when Sokka introduces himself to the Northern Water Tribe's Princess Yue as "Sokka, Southern Water Tribe." The omission is glaring, but Sokka and Katara do not use the full introduction because it requires bringing up their biggest tragedies in life: their father going to war and their mother dying.
Why Sokka And Katara Rarely Mention Their Mother
Sokka and Katara are granted a happy ending with their father Hakoda, as they reunite and join together to help Aang defeat the Fire Nation. Since Hakoda only left them to go to war, Sokka and Katara still held hopes that they would be with their father again, hence why they speak about him liberally with Bato. The same can't be said for their mother, Kya, though. There are several occasions for Katara and Sokka to mention Kya explicitly, such as meeting Hama or at the Northern Water Tribe, but they often fail to do so. Korra never uses this mode of introduction either, yet this is solely because she's the Avatar and in the Water Tribes and needs no introduction.
The reason why Katara and Sokka don't mention their mother is obvious: they're haunted by Kya's death at the hands of the Fire Nation. Especially Katara, who has a vision of her mother in the swamp and hunts down Kya's killer before eventually sparing him. This grief is the reason why Sokka and Katara only mention their mother in response to a relevant event happening, such as when Haru tells her how the Fire Nation attacked and stole his father away. It's also poignant that the last time that Kya is mentioned in The Last Airbender is by Hakoda. After the victory against the Fire Nation, after Hakoda's reunion with his children, and after Katara has got closure after confronting Kya's killer, all three are able to look back fondly on Kya and believe that she's proud of them. And finally, Sokka and Katara have a parent again.