"Modern” is an interesting word. It gets tossed around by cultural commentators, architectural critics, and suspicious traditionalists. Sometimes, it just means “nowadays.” For our purposes here, I’ll define modern as, “Based in a world the reader recognizes as familiar.” So although Moby Dick is certainly a classic, it has a hard time being a modern classic because many of the settings, lifestyles allusions, and even moral codes seem dated to the reader.
A modern classic, then, would have to be a book written after WWI, and probably after WWII. Why? Because those cataclysmic events shifted the way the world sees itself in irreversible ways."
Then that definition wouldn't apply to say, a teenager. Not many teens see the period between WWI and maybe the Vietnam War as being familiar. The cataclysmic event for him might be 9-11. (Just thinking out loud.)