Both responses are binary (hence logistic regression, probit regression can also be used), and more than one response/ dependent variable is involved (hence multivariate).

Is logistic regression a type of multivariate analysis?

Examples of those are multivariate analysis of variance (MANOVA), and structural equation modeling (SEM). The logistic regression is considered like one of them, but, you have to use one dichotomous or polytomous variable as criteria.

What type of analysis is logistic regression?

Like all regression analyses, the logistic regression is a predictive analysis. Logistic regression is used to describe data and to explain the relationship between one dependent binary variable and one or more nominal, ordinal, interval or ratio-level independent variables.