One main theme could be friendship. As Byakuya, who is from a different background, interacts with Naruto and his friends, new bonds are formed. This theme of friendship across different 'worlds' or character origins is quite common.
The theme of cultural clash can also be present. Byakuya comes from his own cultural context within his original story, and Naruto has his own rich ninja culture. The differences and similarities between these two cultures can be explored, such as different values, social hierarchies, and combat ethics.
One common theme is loyalty. Byakuya's loyalty to his family or those he considers family is often explored. Another theme is protection. He is often depicted as a protector in these fanfics.
Another theme is the exploration of their powers in unique situations. For example, some fanfics might have them team up against a new and powerful enemy, and the story focuses on how they combine their abilities to fight. It can be really exciting to see how their different fighting styles work together.
One common theme is friendship. Despite their differences, they often form a deep bond in fanfics. For example, in many stories, they start off as rivals but gradually become friends who support each other.
One common theme is family loyalty. Despite the cold exterior that Byakuya often shows, his loyalty to Rukia as his sister is a central aspect. Another theme is sacrifice. In many of these fanfics, either Byakuya or Rukia is willing to sacrifice for the other's well - being.
One common theme is loyalty. Given their history in the original story, loyalty to each other, whether it's rukia's loyalty to byakuya as her superior or byakuya's loyalty to rukia as someone he cares about deeply, is often explored.
One common theme is loyalty. Byakuya's loyalty to his family, which includes Rukia, is often a driving force in these fics. Another theme is redemption. Byakuya might be redeeming himself for past actions or misunderstandings regarding Rukia.
Love is a very common theme. Their love story, which was complex and filled with sacrifice, is often explored in more depth. Another theme is loss. Since Hisana is no longer alive in the main Bleach storyline, fanfics often deal with Byakuya's sense of loss and how he copes with it.
One common theme is loyalty. Byakuya is bound by his loyalty to the Soul Society and his family, and Rukia has her own sense of loyalty. Their relationship often has to navigate through these loyalties. For example, in many fanfics, Byakuya has to balance his duty to his family's name and his growing feelings for Rukia.
There might be some very niche or creative fanfic writers out there who have come up with this idea. However, it's not a common or mainstream concept. You would probably have to search through various fanfic platforms dedicated to both Naruto and Bleach to see if such a story exists. Maybe on websites like Fanfiction.net or Archive of Our Own.
One of the main themes is elitism. Byakuya Togami, coming from a wealthy and powerful family, embodies this idea. The novels explore how his elitist views interact with the diverse group of characters he encounters.