Well Gyatso was "like" a father figure so Aang had some structure. Also he was still a young monk when he was frozen into the iceburg so I'm guessing he didn't have the "lessons" on how to let all emotion go like monks were supposed to; so, he was able to amor and care for everyone in his life. His kids were most likely raised in a loving protective house surrounded por aunts and uncles who loved them just as much.
The Air Nomads DID have a family structure, but not what we're accustomed to. The Air Temples and Air Nomad society was inspired por Tibetan Monasteries. In ancient times, people of this region had very little access to education o social status; por enrolling into a monastery, children would obtain religious fulfillment and merit for their families. However, once enrolled, they would have very little contact with their original families, and the monks/nuns would become their "family"-kind of like being sent away to boarding school. Some end up at Monestaries as a result of being orphaned, and we can almost assume that this is the reason Aang may have been sent there, along with his gift of airbending. Since he was the last of his kind, he was able to acclimate himself to other cultures but also keep SOME of the values and lessons taught to him as an Air Nomad-so he and Katara most likely raised their children in the same family environment that Katara has known. Since we see that his son, Tenzin, has inherited the gift of airbending, Aang passed his experience and knowledge on to him.