This is an intriguing question. Thanks for bringing it up.
The answer to this question falls into the category of "literary criticism." By nature, the analysis of books can be debated from many different angles; therefore, no answer can be definitive. We can come up with a strong case, so that's what I'll try to help with here.
First, a Bildungsroman is a general classification of story that often gets translated in our culture as "coming of age." The implication is that the main character will grow throughout the story and that the reader will witness this growth. Your question, then, is whether Holden grows enough (or at all.) Why does Catcher in the Rye earn a place in the category of Bildungsroman if its main character is really the same thick-headed chump he was at the beginning of the book?
The answer comes from some subtle nuances of the Bildungsroman. There is an article by a woman named Mairianee Hirsch called “The Novel of Formation as Genre: Between Great Expectations and Lost Illusions.” This came out in a magazine called Genre back in 1979. In it, Hirsch defines some subtle aspects of the Bildungsroman that might help clarify the situation with TCITR.
- The "Bild" is, at heart, a quest story in which the main character attempts to learn more about life within a structured society (Holden meets this criteria.)
- The character should be rudely disconnected from their previous, stable life (the death of Holden's brother and his subsequent assignment to various boarding schools.)
- The character's growth is slow and the product of repeated clashes with society's norms (Holden is always bumping up against what everyone else is doing.)
- The character, by the end, has a better understanding of himself and his place in the society that he's been smashing up against (this is the one that really makes your question interesting...has Holden learned anything by the end?)
In short, Catcher in the Rye may often be reffered to as a Bildungsroman for one of two reasons:
- It is easy to just call it that. It's about a kid, life, and a journey. People who don't want to think much about the story can easily label it as "coming of age."
- The essence of the Bildungsroman is not in the measure of how great the change is, but rather the idea of a young character trying to find his or her place in a world they don't seem well suited to. The character continually butts up against societal norms and either changes or cracks up (another type of change, really.)