Azula is the family member Zuko cares the least about, by far [and no, it's not because she's mean to him and no, this is not a post about Zuko's morality or anything like that]
So the entire premise of Zuko's arc is he spends years and years trying to regain his father's love and appreciation, that he spent three years trying to capture the Avatar so that his father would like him again. We even see in a flashback that exiled Zuko misses his father badly.
And when Zuko decides to completely break off his relationship with his father, he decides that this merits a dramatic confrontation, since his relationship with Ozai is so important to him. Even after that, the second to last scene of the series is Zuko visiting Ozai.
Zuko often takes Iroh and his support for granted, but even in Books 1 and 2 there are plenty of moments which act to reaffirm how much Zuko values and appreciates his uncle. He goes out of his way to protect his uncle on a couple occasions. And in Book 3, after Zuko has screwed up his relationship with Iroh, one of Zuko's core motivations becomes fixing his relationship with his uncle and regaining Iroh's love, trust, and appreciation.
Ursa is someone who Zuko talks about far less, yet the series makes it very clear how much Zuko misses her, with "Zuko Alone" and "The Earth King" and "The Day of Black Sun: The Eclipse." And the second to last scene in the series is Zuko trying to find her.
That leaves Azula. Zuko never seems to miss Azula the least, even though he hasn't seen her for years when the series begins. In fact, when she unexpectedly shows up and greets him, instead of being happy to see her, he's angry and suspicious (to be fair to Zuko, she did turn out to have ulterior motives due to Ozai's orders, but if he really deeply missed her, he'd probably be at least a little happy to see her regardless. Mai and Ty Lee reacted with joy the first time they saw Azula again). And in Books 2 and 3, there is essentially nothing to suggest that Zuko misses the better relationship he had with Azula when they were younger.
For instance, in "The Beach," Zuko angsts over an old photograph of his family.
and he remembers a happy memory associated with Ozai:
and then Zuko's eyes focus on Ursa
and he remembers a happy memory associated with her
Yet Zuko never looks at younger Azula in that picture, nor does he think of her at all. The "Happy Family" that Zuko misses includes Ozai, Ursa, and even Iroh and Lu Ten (shown in a separate memory sequence at little earlier), but not little Azula.
Ultimately, what we see in the first half of Book 3 is that Zuko is willing to take advantage of Azula's kindness, when she shows it to him, but he never acknowledges it, thanks her for it, or reciprocates in the slightest.
In the end, in the DoBS Zuko leaves Azula behind without a regret or even a thought. Ozai gets a big confrontation; Azula gets nothing. And after that, we see that Zuko misses Mai, and still feels complex feelings for Ozai ("Fatherlord," anyone?), but there's nothing to suggest he misses Azula at all or feels a deep attachment to her. The penultimate scene of the show even features Zuko visiting Ozai, not Azula.
Some people might point out that Azula is a bad sister to Zuko. That's definitely true. She says and does plenty of awful things to him over the series.
Yet that doesn't explain things. Ozai is far, far worse to Zuko than Azula ever is, yet Zuko cares deeply about Ozai to the degree that Zuko's entire arc is about it. Merely talking about "look how mean Azula is" explains almost nothing. Is possible to explain why
This does bring me to my main point. The ATLA fandom tends to believe that Zuko cares about Azula a lot and that he places a lot of value in having a good relationship with her, tends to believe that Zuko is obsessed with Azula.
However, the reality is that Zuko by far cares about Azula the least of any member of his family and that he places almost no value on having a good relationship with her. She is never a priority for him and is at most only an obstacle. All of Zuko's other family members loom far higher in Zuko's priorities and headspace than Azula. And if he is obsessed with her, it's not in a loving way. Maybe "Azula the enemy" and "Azula my advisor and emotional caretaker" are significant to him, but "Azula my sister" never seems to be.
Again, this isn't a moral judgement, merely a fact.