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In my textbook, it is written that the binomial name of mango is Mangifera indica and binomial name of bee is Apis indica. Now in the name the second part is the name of species. But mango and bee are not same species. One is tree and another is animal. Then why is their second name same?

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 nomeclatural 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.

That is the species name it is often the same for unrelated organisms, that is why we use a two name system. Binomial nomenclature goes Genus species

the first identifies the genus (which should not be the same unless they are closely related) and the second is the species name which is often little more than a descriptor, you would be amazed how many species names translate to simple concepts like colors or one of the most common, familiaris which just means household or common. In dinosaurs many many end with carnegii because Andrew Carnegie funded so much paleontology and scientists knew who buttered their bread. In your case indica literally means "from India" in latin so it is not surprising many things carry it.

Mangifera indica means 'mango from India', Apis indica means 'honeybee from India', guess where they were first discovered.

In short the second name really doesn't matter as long as it does not match anything else with the same genus name. As long as the combination of the two names is different and unique. the first name, the Genus name, however should not be the same unless they are related.