War Drama
This one should be pretty self-evident. One thing to keep in mind, though, is that this book definitely isn't a glorification of war. War dramas don't necessarily need to be pro-war (by which we mean supportive of a particular war, not just gung-ho about war in general), and definitely some of the more famous war dramas are not: consider books like All Quiet on the Western Front and The Things They Carried. A book's attitude or stance toward war is perhaps the most important thing to consider in determining what kind of war drama it is.
Coming-Of-Age
You may not think it at first, but Johnny Got His Gun is also a coming-of-age story. You might say, though, that it's a coming-of-age story that gets disrupted before it gets to adulthood.
What makes this novel a coming-of-age story is that we get different stages in Joe's development and adolescence, and they are things that most teenagers go through in some form or another: girlfriend drama, friend drama, family drama, all kinds of drama. Oh, yeah: there's also war drama. What, did you think this was The Catcher in the Rye?
War is what shakes up this story as a regular coming-of-age tale. Joe tells us that he went to war without really understanding "what the fight was all about" (2.22). Now, in a regular coming-of-age story, this kind of naivety might be mildly punished—you know, just enough for Joe to learn his lesson and grow up a little. In this novel, however, our hero's naivety gets the crap beaten out of it.
We might consider something Joe thinks on the day that he leaves for the war:
Oh Kareen why do they have a war right now just when we find each other? Kareen we've got more important things than war. Us Kareen you and me in a house. (3.40)
Joe's first real experience of romantic love is rudely interrupted, and as we know, it's not going to get the chance to recover. Joe doesn't get to learn and grow from his experiences; instead, his movement toward adulthood is halted, and Joe gets suspended in this strange limbo where there is no real future for him. So maybe instead of coming-of-age, we might call it coming-of-the-worst-reality-you-could-possibly-imagine.