The regression package brms (for R) supports multivariate regressions models. Building on the generalized linear models framework, the response variables may even be on different scales (e.g. metric, counts binary). There is a vignette here:
In using multiple regression modeling, the relationship between one variable and 10 other variables should depend on the type of the single variable. if it is binary, multiple logistic regression is recommended. if the variable is quantitative, multiple linear regression is recommended.