Q: Is Oily skin actually a type of skin you can have, or the body's reaction to not drinking enough water / using drying skin products?
A: This is a fantastic question, because it gets to the core of what skin care actually is: care for an organ of the body. In short, yes, certain skins can just produce more oil than others without any special circumstances. This high oil production can exist independently of any other apparent condition in the body, as people come in all colors, all sizes, and all skin oilinesses.
It can also be that the skin is responding to circumstances inside the body by either over-producing or under-producing sebum. Typically this relates to inflammation--I talk about this at length in my video Ask an Esthetician: Acne, Hyperpigmentation and Eczema Together? (http://www.youtube.com/watch?v=qvz8itBAEMA). The skin is very sensitive to inflammation in the digestive tract of the body in particular, and it's not at all uncommon to see chronic inflammatory conditions of the skin like acne or eczema in people who have poor digestive health.
As for over-stripping the oils of the skin, the skin responds best to a cleansing routine that removes only enough of the natural sebum to prevent comedogenic issues (ie, follicular blockages that lead to acne), but not so much that the skin's lipid layer becomes compromised. Naturally oily skin is reputed to visually age more slowly due in part to its superior ability to retain water in the epidermis (see: TEWL), and that ability is only as good as the lipid layer is strong. So the goal for all skins, not just oily types, is to only use cleansers that are effective enough to regulate the natural environment, but not so harsh that the environment becomes unnaturally compromised.
Even using all of the right products, even in perfect health, it's still possible for individuals to have oilier skin than average. However it's possible to compromise the health of the skin both intrinsically (due to internal circumstances in the body) and extrinsically (due to external circumstances), leading to inflammatory symptoms of various types of acne, eczema, and irritation.
Re: water consumption, there is no direct link I'm aware of between water consumption and oil production. Drinking water is good for your digestion and metabolism, though, and your skin is sensitive to both.