I used to believe that the negative binomial distribution (for count data) and Gamma distribution (for continuous data) shared the property that the variance can take arbitrary values regardless of the mean, which makes them quite flexible as opposed to the Poisson distribution.
However I've recently seen in a variety of places that statisticians consider the variance to be (proportional to) the square of the mean (the Wikipedia Variance function page even plainly claims that $V=E^2$ by the way). To me, this seems to be vacuously true just like any two non-zero real numbers are proportional to each other.
The main point that I'm trying to make is that when you use a GLM based on the Gamma or negative binomial families, the coefficient of proportionality (the shape parameter) can vary with the IV, it doesn't have to be constant.
What am I missing?