Why do both the mango and the bee have "Indica" in their binomial name?
In short, we do not think about the uniqueness of the second part of the binomial (the species epithet) but about the uniqueness of the binomial itself (the genus and the species epithet). Thus, the unique binomial of mango is Mangifera indica and the unique binomial of bee is Apis indica. For more detail, see this question.

To complicate matters  slightly, plants and animals are governed by different nomenclatural codes. So it is possible for a plant to have the exact same binomial as an animal. These are called "hemihomonyms." For more detail, see this question. However, a plant cannot have the same binomial as another plant, and an animal cannot have the same binomial as another animal. 

In this specific case, the authors probably chose to give both species the epithet indica because they are associated with the Indian subcontinent, which is the root of that word.