To start, Azula's mom didn't seem to like her. Though I'm sure Ursa did, Azula just didn't think so. And that's what matters. As a parent, you need to make sure your child knows you love her/him. Ursa neglected Azula because she thought Ozai's respect for her was enough. And because she thought Zuko needed her love more. Which was not true, Azula needed real love, not just her father's appreciation for her power. I'm not saying Ursa is a horrible mother, she wasn't abusive or anything but she was rather negligent. Ursa also disciplined Azula and yelled at her until she felt as though Ursa thought she was a monster. When parenting the point of discipline isn't to make the child feel like a bad kid it's to teach them not to do whatever they did again. Ursa's method of discipline made Azula feel unloved. Not to mention that she pretty much stated to little Azula's face "what's wrong with this child?" Sure she didn't act like it hurt her but you can tell it did, especially in the end. It really says a lot that the girl had a hallucination of her own mother...telling her (possibly) what she wanted to hear.
As of Ozai...this buttmuncher essentially used her. She was like a pawn to him. A lot of people seem to glamorize her relationship with him. But honestly it really wasn't much better than Zuko's. Sure there was not physically abused...but the a sort of mental abuse was evident. Azula just wanted her father to love and appreciate her. Hell, she wanted his respect as much as Zuko did and I think that's why she did so many of the things she did throughout the series...to gain his affection and impress him. And possibly to avoid getting hurt as Zuko was. And so she did earn his respect but not genuine love. Which is sad, parental love should be unconditional, she (nor Zuko) should have to 'prove their worth.' When it came down to it, Azula was simply being used. And by her own father. He had no problem casting her aside when he thought he was going to get more power and had no use for her. Sure he gave her the fire lord title she wanted, but only when he thought he would be the phoenix king. The crown was essentially useless at that point, and he knew it too. The look of hurt on her face when he yelled at her was also rather painful. I also read this book from the points of view of each of the characters; at that part of the book she mentions that she's afraid he's going to hurt her as he did Zuko. So really, when it comes down to it, Azula's relationship with Ozai is no better than Zuko's.
Long story short, I find it hard to fathom how people don't seem to think Azula was abused in some way shape or form. Negligence is technically abuse. And the sort of unspoken fear Azula had of Ozai says a lot.
As for her relationship with Mai and Ty-Lee; her interactions with them seemed less friendly and more political (I'll explain that in the next paragraph). Though I do think she liked them, she just didn't know how to show it properly. She was never really taught how to show affection right. And in the end this is what leads to her losing them. And again it hurt her more than she let on--and I think she actually did express that it hurt her. Her hurt came out in the form of anger "put them somewhere where I'll never have to see them again and let them rot." Speaking from experience, when someone hurts my feelings I do the same thing I don't cry, I try to make them feel as bad as me, especially if I care about them. It's odd but and I don't quite know how to describe it...it's like hating someone you love for not loving you. She really shows how much it hurts her when she mentions that the Di Lee agents would betray her just like Mai and Ty-Lee did. Where as Zuko and Aang where surrounded by friends. Zuko always had Iroh, he just didn't seem to see it. Azula is completely alone. I know that Grey DeLisle mentioned in an interview that Azula used fear to make people be her friends because she didn't think people would like her because they wanted to. That in itself is rather heart wrenching. And that's without adding the fact that the fear she thinks she has to make them feel is what costs her possible friends in the end. It's like a vicious cycle.
As shown on the beach, Azula as some...social troubles. Yes she's good at manipulating them and reading them, but she doesn't know how to talk with them or make friends with them. And that in itself is sad; the fact that she's so well-versed in political social skills yet lacks any other form of it. It shows that she was probably put in a position where she was really only able to communicate with people on a political basis. Not to mention her mother never showed her love, which in turn didn't show her how to love. And Ozai certainly wasn't a good example--in fact I feel like he was the one she modeled her way of showing affection after. I mean he showed her affection in a more political way, so to speak, and there for her affection was more based on respect than genuine love. I would also think that her 'privet and esteemed' lessons at the Fire Academy for girls didn't help much either, as that still seems like a more political learning environment. Basically Azula was raised for politic through and through, and she received little to no actual social skill set training.
