The drag equation uses v2 because the force depends on the square of the speed. If you go twice as fast, the drag is 4 times bigger. That's what has been observed to hold. You don't use v because it doesn't depend linearly on v. There is probably a theory that shows why the term is v2, but if anyone here knew it they'd probably have posted it by now.
If I had to guess, I'd say that it's a combination of imparting momentum to air molecules by collision, which depends on the relative speed, and trying to compress/do work on the air (or whatever fluid) which would also depend on the relative speed. So you end up with a combination of the two effects, which is nonlinear and thus depends on v2. Again, this is a guess.