Definitely not. The Black Butler series has a set of established themes such as the supernatural elements, the Victorian - era setting, and the complex character relationships. The idea of a 'leashed slave story' is completely foreign to the official canon and is likely a disturbing creation outside of the proper series. It would go against the overall tone and moral framework of the official work.
There is indeed a 'Black Butler' visual novel. It features the same dark and alluring art style as the anime and manga. The visual novel might include different storylines, some of which could be exclusive to this format, providing a new perspective on the 'Black Butler' world. It often has a strong focus on character development and the complex relationships within the story, which is great for fans who want to explore more about the characters.