It is a common misconception that logistic regression does classification just because the $y$ variable is discrete. Logistic regression estimates the conditional probability of an event, and all $p\in(0,1)$ are fair game (arguably all $p\in[0,1]$). If one desires, one can apply a threshold to categorize the probability outputs, but this is not necessary, and there’s certainly no notion of “misclassification” in the logistic regression on its own.
As is typical of generalized linear models, the parameters of a logistic regression equation are estimated via maximum likelihood, considering the conditional response variable as Bernoulli.$^{\dagger}$ A GLM textbook like Agrest’s Foundations of Linear and Generalized Linear Models gets into the details.
Agresti, Alan. Foundations of linear and generalized linear models. John Wiley & Sons, 2015.
Vanderbilt professor Frank Harrell is a major opponent of logistic regression thresholds. Two of his blog posts get into why.
https://www.fharrell.com/post/class-damage/
https://www.fharrell.com/post/classification/
$^{\dagger}$It’s possible to consider the conditional distribution to be a binomial with more than one trial (coin flip), but this is not what you mean when you want to use logistic regression to distinguish between pictures of dogs and cats (for example). That is a binomial distribution with one trial, which is Bernoulli.
