One common theme could be power struggle. Since Voldemort is all about power and Harry is his main obstacle, when Voldemort 'breaks' Harry, it shows a shift in the power dynamic.
Perhaps Voldemort has some inside help. He could have bribed or threatened one of the guards or even a Ministry official who has access to Azkaban. This person could then help him get in and break Harry out. Or he might have found a secret passage into Azkaban that the Ministry doesn't even know about, and he uses that to his advantage.
There are likely many 'voldemort breaks harry fanfic' out there. Some might explore what would happen if Voldemort managed to physically break Harry in a fight, perhaps leaving him severely injured and how that would change the course of the wizarding world. Others could be more metaphorical, like breaking Harry's spirit or resolve in the face of Voldemort's evil.
One reason could be that Voldemort wants to use Harry for his own gain. Maybe he thinks he can turn Harry to the dark side and use his power for some grand evil plan. He might believe that with Harry on his side, he can finally defeat all his enemies and rule the wizarding world completely.
I'm not sure if there is such a fanfic. Fanfics cover a wide range of themes and scenarios, but this particular one might be quite rare or even violate some of the general guidelines of the Harry Potter fanfic community. Many fanfic platforms have rules against inappropriate or non - canon - compliant content.
There probably is. Fanfic communities are full of unique and unexpected plot ideas. This kind of plot would completely turn the original story on its head. It might involve some sort of alternate reality where Harry and Voldemort have a common goal or where they are forced to work together due to some external threat. It could be a really interesting read for those who like exploring different interpretations of the Harry Potter universe.
At first, it would be extremely tense and full of suspicion. Harry would be very wary of Voldemort's motives for protecting him. Voldemort, on the other hand, might be impatient with Harry's lack of trust.
Perhaps Voldemort discovers that there is a greater threat out there. In the fanfic, it could be that another dark force is emerging, and Voldemort sees Harry as an ally of sorts against this new threat. So, he decides to protect Harry in order to deal with this common enemy.
Snape could also be a character in such a fanfic. His complex relationship with both Harry and Voldemort makes him an interesting addition. He might be in a difficult position, having to keep his cover as a double - agent while also trying to find a way to save Harry without revealing his true intentions to Voldemort.
At first, it would probably be very tense. Harry would be suspicious of Voldemort's motives, and Voldemort would be cautious of Harry's potential to rebel. But as time goes on, they might develop a strange sort of understanding. Harry could start to see Voldemort as more than just a pure evil figure, and Voldemort might start to view Harry as something more than an enemy.