The premise is this quote from vignette of R package
Further-more, the model shares some properties (such as linear predictor, link function, dispersion parameter) with generalized linear models (GLMs; McCullagh and Nelder 1989), but it is not a special case of this framework (not even for fixed dispersion)
This answer also makes allusion to the fact:
[...] This is a type of regression model that is appropriate when the response variable is distributed as Beta. You can think of it as analogous to a generalized linear model. It's exactly what you are looking for [...] (emphasis mine)
Question title says it all: why Beta/Dirichlet Regression are not considered Generalized Linear Models (are they not)?
As far as I know, the Generalized Linear Model defines models built on the expectation of their dependent variables conditional on the independent ones.
$f$ is the link function that maps the expectation, $g$ is probability distribution, $Y$ the outcomes and $X$ the predictiors, $\beta$ are linear parameters and $\sigma^2$ the variance.
$$f\left(\mathbb E\left(Y\mid X\right)\right) \sim g(\beta X, I\sigma^2)$$
Different GLMs impose (or relax) the relationship between the mean and the variance, but $g$ must be a probability distribution in the exponential family, a desirable property which should improve robustness of the estimation if I recall correctly. The Beta and Dirichlet distributions are part of the exponential family, though, so I'm out of ideas.
 Cribari-Neto, F., & Zeileis, A. (2009). Beta regression in R.