The Non-Gaussian factor model introduces the idea of using other distributions for a factor analysis. But that code is not very generalized, and in reality we’ll tend to need something more like structural equation modeling with non-Gaussian variables.
The name for a factor model with logit-linked indicators, whether dichotomous or ordinal, is Item Response Theory. It has been used for decades to develop instruments and in particular, tests. Because of its history, factor loadings are called the “discrimination” parameters, intercepts are the item “difficulty”, and the factor scores represent each person’s “ability”.