I think that you can definitely call this book a bildungsroman because it follows the title character as he grows and matures and it focuses on this process of change that he undergoes.
A bildungsroman, we are told in the "bildungsroman" link,
intends to lead the reader to greater personal enrichment as the protagonist journeys from youth to psychological or emotional maturity. Traditionally, this growth occurs according to a pattern: the sensitive, intelligent protagonist leaves home, undergoes stages of conflict and growth, is tested by crises and love affairs, then finally finds the best place to use his/her unique talents.
This is exactly what this novel does.
Over the course of the story, we follow Kim as he (literally and figuratively) finds out who he is. We start with him as an orphan who does not even know what race he is. As we go along, Kim discovers who he is in a literal sense -- he finds out who his father was and what regiment he was in.
Because of this discovery, Kim ends up finding himself in a more figurative way. He is sent off to school and gains not only book learning but also instruction in how to act as a spy in the "Great Game." By the end of the book, he has found a place for himself. He has found a profession and he has also found a sense of belonging.
By doing these things, Kim develops himself and his potential. This, to me, is the very definition of a bildungsroman.