What is the flavor quality that makes cheese "sharp"? Are there other seasonings or foods with this "sharp flavor?

Perhaps you are referring to the umami flavor.

Per cooks illustrated:

There’s more than one reason that classic combinations like a burger and cheddar cheese or the Parmesan cheese and anchovies in a Caesar salad taste so good: Not only do such ingredients simply go well together but it turns out that they contain complementary umami boosters that magnify flavor by as much as thirtyfold. More and more food scientists are concluding that when ingredients rich in naturally occurring glutamates are paired with ingredients that contain either one of the nucleotides inosinate or guanylate, the perception of umami, or savoriness, is dramatically

Rich In Glutamates

(MG/100 G)

• Parmesan cheese 1,200-1,600
• Fish Sauce 950-1,383
• Soy sauce 800-1,300
• Tomato paste 680
• Cured ham 337
• Anchovies/sardines 280
• Beef 107
• Cheddar cheese 78
• Worcestershire sauce 34

Rich In Nucleotides

(MG/100 G)

• Anchovies/sardines 193 (inosinate)
• Dried shiitake mushrooms 150 (guanylate)
• Pork 122 (inosinate)
• Beef 107 (inosinate)
• Dried porcini mushrooms 10 (guanylate)

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.