What is the flavor quality that makes cheese "sharp"? Are there other seasonings or foods with this "sharp flavor?
The term/process you are looking for is acidity or acidification.  During the cheese making process lactose is broken down into lactic acid.  This lowers the pH and creates the tart, sour, or acidic taste you are referring to. Higher acid cheeses (lower pH) present as "sharper", an old cheddar, for example.  Ever have a salt and vinegar potato chip?  I get a similar flavor perception (minus the lactic, cheese flavors of course).  So, other acidic ingredients (vinegar, lemon) can easily get you in the same flavor perception ball park.