TAYLOR: It's scary 'cause nobody tells me how to do anything 'cause I'm new. Nobody cares about the new guys. They don't even want to know your name. The unwritten rule is a new guys' life isn't worth as much 'cause he hasn't put his time in yet.
Some family we've got here. This is a special family alright; one where you have to earn your way in. You're automatically family in the platoon because if you're knew, well, "nobody cares" about you.
TAYLOR: Of course mom and dad didn't want me to come here. They wanted me to be just like them. Respectable, hard-working… they drove me crazy with their goddamned world.
Taylor clearly isn't a huge fan of his parents. He doesn't want "their world," and he definitely doesn't want to be like them. You could even say he goes to Vietnam in search of a new family.
TAYLOR: Barnes was at the eye of our rage, and through him, our captain Ahab, he would set things right again, that day we loved him.
The mention of Captain Ahab here (legendary captain from Herman Melville's Moby Dick) suggests that the platoon is kind of family, a group of guys that, just like the sailors in Melville's novel, spend their lives together with their patriarch, Barnes-Ahab.
TAYLOR: A civil war in the platoon, half the men with Elias, half with Barnes… I can't believe we're fighting each other.
A little family quarrel is expected from time to time, but the family feud here is definitely imbued with much more serious consequences. We're not talking about a brother who won't talk to his sister, but guys that are actually at war—"fighting with"—with each other instead of the enemy.
TAYLOR: The war is over for me now, but it will always be there, the rest of my days, as I'm sure Elias will be, fighting with Barnes for what Rhah called possession of my soul. There are times since when I have felt like a child born of those two fathers…
Taylor is very much the child of Barnes and Elias, his two "fathers." Taylor shares Elias' overall perspective, but he finishes his tour in Vietnam by doing something Barnes would have done (he kills Barnes, much as Barnes had killed Elias). Although Taylor kills one of his fathers (Barnes), that fathers' influence persists in the act itself.