Why is cooking in oil or fat considered dry heating, while cooking in wine is considered moist heating?
Oil is dry heat because oil contains no water. Wine does.
The "moist" in moist heat really means water. In moist heat cooking, water acts as a solvent and actually dissolves much of the solid matter in the food - hence the reason why steaming and boiling tend to make food rather soft or even soggy.
Oil, on the other hand, is very rarely a solvent. There are certain food compounds that are fat-soluble, but most are water-soluble and oil, being hydrophobic, actually blocks any contact with water, thus preventing any dissolution, and generally making the food crispier as well as promoting the Maillard reaction (which can only occur with dry heat, due to the low boiling point of water).
It may seem counterintuitive, but oil isn't wet. It just looks that way.
To build on Aaronut's answer, one issue with cooking in a "moist" environment is that it limits the temperature.
In dry-heat cooking, such as roasting, searing, frying, grilling, etc., the flavor from the Maillard reaction (to which Aaronut alluded) is a very important result.
In moist cooking, such as braising, poaching, steaming, etc., the temperature cannot (with the exception of pressure cooking, but there are other issues there) rise high enough to allow your meat to become browned. You can also have flavor and nutrient loss, as well as texture issues.
Consider a steak, for example. Whether it is pan-seared or grilled, it will be browned on the outside. Imagine if, instead, it was boiled. Personally, I would not want to eat that gray lifeless mass of meat.
Or another example: french fries. You can put potatoes in hot oil, and you get a crispy delicious snack. Drop them in boiling water, and you get.. boiled potatoes?
This isn't to say there is no value in moist cooking; just to illustrate the difference between moist and dry.