Does this formula hold for the coefficients of a logistic regression $\pmb{\hat{\beta}} \sim N(\pmb{\beta}, (\mathbf{X}^T\mathbf{X})^{-1}\sigma^2)$? A person on Cross Validated states that the coefficients of the general linear model follows the following distribution
$$\pmb{\hat{\beta}} \sim N(\pmb{\beta}, (\mathbf{X}^T\mathbf{X})^{-1}\sigma^2)$$
Does this still hold for logistic regression?
Same question for probit regression.

No. In general, the coefficient estimates in logistic and probit$^{\dagger}$ regressions are biased. In the distribution given, the expected value of $\hat\beta$ is $\beta$ itself, meaning that $\hat\beta$ is unbiased.
Further, it is not clear what $\sigma^2$ would be in a logistic or probit regression. The given formula implies a constant conditional variance (which is $\sigma$), and the variance of a binomial distribution depends on the probability parameter that is being estimated by the regression. Since that probability is unlikely to be constant, the conditional variance is unlikely to be constant, and $\sigma^2$ is ambiguous.
$^{\dagger}$Use an argument similar to the one in the link about the nonlinear function.
