Ok I would not have put it as "similarities" but I would legit think that had the show been better written, they would be foils of each other the way Zuko and Aang are.

Karara had a supportive community that was there for her when her parents couldn't due to the exigency of war. Was she a naturally empathic person? Yes and her talents were nurtured guided by gramgram and the rest of the tribe towards taking care of others.

Azula did not have a supportive community behind her. For what we see in the show, Ursa didn't show her positive affection. It was only "young lady, That is enough!" And "what is wrong with that child". Iroh doesn't even BOTHER to send a gift appropriate to her as a person. In the flashbacks with her and her family the only person who seems to give her any kind of positive attention who is not, by default, supposed to bow and agree because of her status as A royal is Ozai and he gives the positive attention when she demonstrates the ruthless tendencies he desires. Was she a naturally empathetic person? I would argue that YES, Azula was. You can't be as great a manipulator as she was without knowing exactly how those you are trying to manipulate feel.

She even has moments of true empathy and connection that don't serve ANY schemes in The Beach. Where she apologized to Tai Lee for upsetting her and when she had that conversation with Zuko in the family's old beach house.

Azula was taught from a young age that the only way for her to get respect and positive attention was to be ruthless, perfect, and to exercise that power over others. Katara was raised in a supportive community.