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In GLM, it seems logistic regression is related to Bernoulli Distribution. How are they related mathematically?

Tinyik
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Logistic regression assumes the response is conditionally Bernoulli distributed, given the values of the features.

$$ y \mid X \sim Bernoulli \left( p = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}} \right) $$

The Bernoulli distribution has one parameter, the probability of the positive class. Logistic regression also specifies a specific functional form for this probability in terms of the features.

Matthew Drury
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Logistic regression models the mean of a Bernoulli. The logit link function of the mean is the linear function fit by logistic regression. The only additional requirement for GLM is that the domain of the variable is restricted to 0 and 1