Holden is obviously a rebellious adolescent with a genius I.Q. He may make a lot of mistakes in life, but he insists on thinking for himself and not accepting other people's opinions unless they agree with his own experience and common sense. He doesn't agree with Dr. Thurmer that life is a game. (Neither do many of the readers of The Catcher in the Rye!) Life is not a game and Holden knows it. Dr. Thurmer probably knows it himself and is just saying it to be saying something. Furthermore, Holden doesn't see why he himself should abide by anybody else's rules. Since any school has to have rules and regulations, Dr. Thurmer may be trying to justify the many rules at Pencey by using the analogy of a game. But are the school rules at Pencey intended to make the students feel that they are engaged in a game like softball or soccer? The rules are serious and are intended to enforce discipline. Holden is aware of that. It is obvious. And he has no particular objection to the school's having a lot of rules. But he doesn't like to be talked to as if he were an idiot and could be persuaded that he was just playing a game by obeying all the rules. He likes to test reality. He likes to see what happens if you don't obey the rules. This is something Dr. Thurmer finds upsetting. The fact that Holden is being kicked out of Pencey for not playing by the rules shows that the rules are very serious indeed.
The significance of this comment is that it reflects the kind of adult world view about abiding by social rules and conventions, which Holden, in the full throes of adolescence, rebels against, at least in his thoughts. While agreeing outwardly with the statement, as repeated by Mr Spencer, Holden remarks in a cynical aside to the reader:
Game, my ass. Some game. If you get on the side where all the hot-shots are, then it’s a game, all right – I’ll admit that. But if you get on the other side, where there aren’t any hotshots, then what’s a game about it? Nothing. No game. (chapter 2)
Holden resents this labelling of life as a mere ‘game’, as though it’s easy for everyone to achieve happiness and success simply by following a designated set of rules. Lost, confused, repelled by his peers and grown-ups alike, he feels things are much more complicated than that. To him, adults are really trivialising matters by calling life a game. Maybe it’s okay for the ‘hotshots’, the popular, good-looking, well-connected types that Holden feels inferior to, but other people, he thinks, don’t stand a chance.