Why is the normal distribution family in GLMs homoscedastic? As the question asks, what allows us to assume that the normal distribution family exhibits homoscedasticity?
 A: The normal family of distributions for a regression model $Y_i \sim \text{N}(\mu_i=x_i^T \beta,\sigma^2)$.  This means that we assumes that our random variable $Y_i$ have some normal distribution with some unknown mean $\mu_i=x_i^T \beta$ and unknown variance $\sigma^2$.  This says that, whatever is the value of the predictor variables $x_i$, the variance of the observation is the same. That is homoskedasticity!
We could have written the model in some other way, such as (lets say with one predictor variable $z$) the distribution is $Y_i \sim \text{N}(\mu_i=\beta_0+\beta_1 z,\exp(\gamma z))$, where the variance is some function of the predictor $z$, depending on parameters to be estimated.  Such a model would be heteroskedastic, but is not a part of the usual glm framework.  In R such models can be estimated with the gamlss package (and certainly others).
 EDIT

Answer to extra question in comment:  Look at the binomial family. For the normal family, we have two independent parameters, so we can specify mean and variance separately.  Not so for the binomial model! If $Y \sim \text{Bin}(n,p)$, where $n$ is not considered usually a parameter, since it is defined by the data, we have only one parameter $p$. Then we can calculate that the expectation is $\mu=n p$ and the variance is $\sigma^2= n p (1-p)$. Note that we can write the variance as a function of the expectation: $\sigma^2 = \mu (1-\frac{\mu}{n})$. So, in say, logistic regression, when we estimate in some way the expectation as a function of covariables, that will determine automatically the variance as a function of covariables.  There is no freedom in estimating the variance as soon as we have estimated the mean, is it already determined.
 The same happens for the Poisson family. 
