Unfortunately schools often fail to teach the message behind the works we view as classics.
Shakespeare wrote not just to entertain, but to inform and comment on what was happening at the time. Unfortunately many teachers don't do a good job, or don't have the time, to explain the background of the works. Students often don't know or understand that his works were often directed at specific royalty or other influential people. And so when we first read his works, we don't get the insults, slights, and compliments that would have been obvious to people back then.
Other classics speak about what it is to be human, and evoke imagery that most works never manage. What we consider to be the classics are only a small percentage of novels, poetry, and essays created in their time. There are works being written today that will become classics, but they have to be shown to be meaningful to future generations, just like the current classics have.
There are some games telling wonderful stories, and maybe they'll come to be regarded as classics in their own right. But the Super Marios, Warcrafts, and even Assassin's Creeds are not going to be among them. They're pioneers in their own right, but they don't actually have anything to teach us about the human condition, politics of their day, or have a timelessness that help towards enlightenment.