My first thought when I saw that scene was ''Azula must be bad at feeding turtleducks and Zuko was so overexcited about the fact that he was better than her at something that he didn't think about the consequences of replicating her humiliation."
I don't think I had a clearly articulated thought process at the time, but my reasoning boils down to the fact that Zuko's behavior resembles mockery more than moral condemnation. People do not usually express "this person is cruel" by replicating their actions. When you think something is wrong and immoral you do not usually want to behave that way yourself. However, it is very normal to mimic someone's behavior in an exaggerated manner when you think what they did was silly or stupid.
By calling out "Hey mom, do you want to see how Azula feeds turtleducks?" Zuko was signalling that he was starting a performance and inviting Ursa in on a joke at Azula's expense. It is a joke because he thinks that how Azula feeds turtleducks is incorrect in some way and that he and Ursa, who feed turtleducks correctly, should now laugh at her failure. This is the behavior modeled to him in a family largely made up of bullies. The childish logic at play here is "Azula always treats me like this when I fail at bending, so why shouldn't I make fun of her when she fails at feeding turtleducks?"
At the same time, his willingness to replicate the event suggests that he considered it more entertaining than immoral. Once Zuko hits the turtleduck, he seems genuinely surprised to realize he had hurt it. He is shocked and confused when it attacked him, demanding to know why it would do that. This is important because it shows that Zuko didn't understand that his actions would cause the turtleduck to lash out in distress, which suggests that he didn't consider Azula's actions to be harmful either. His only take away was how funny it was too see Azula mess up at something as simple as throwing bread in a pond.
Zuko's specific confusion at the turtleduck's biting him is also telling because it suggests that the last turtleduck hadn't attacked Azula. We can't say why this is for sure. Maybe she hit the mum rather than the baby and it didn't feel the need to go on the defense, or maybe she didn't hit the turtleduck at all and just made a huge splash when she threw the bread too hard. Perhaps she was told to "throw the bread into the water" and she threw it like she was taught in combat training, making a commotion but not actually hitting the turtleduck. Whatever she might have done, the fact that she wasn't attacked suggests that her actions only disgruntled the turtleducks rather than make them feel threatened the way that Zuko did.
And if Azula hadn't made the turtleducks feel threatened, it is probably because she wasn't trying to (intimidation is one thing we know she is good at). She was probably trying to feed turtleducks as people normally do and messed it up in someway that Zuko found funny. In trying to replicate her mistake for the sake of humor, Zuko chose to intentionally aim at the turtleduck itself without thinking of what the consequences of that would be.
This is mockery. He isn't judging Azula on the grounds that her actions were wrong, he is making fun of her because he thinks what she did was embarrassing. The sentiment he is trying to share with Ursa is "Azula cannot feed turtleducks normally, how do you mess up something so simple? It looks ridiculous!" His inability to predict the events that followed (hurting the turtleducks, being attacked back) suggests that those things hadn't happened the first time.
I do not think that either child was trying to hurt any of the animals. However, after years of constantly falling behind his little sister and having her make fun of him for it, it makes total sense to me that Zuko would jump at the chance to make fun of Azula for something that she is bad it without thinking about what he was doing.
