This study focuses on mixed-variable black-box optimization (MV-BBO), addressing continuous, integer, and categorical variables. Many real-world MV-BBO problems involve dependencies among these different types of variables, requiring efficient methods to optimize them simultaneously. Recently, stochastic optimization methods leveraging the mechanism of the covariance matrix adaptation evolution strategy have shown promising results in mixed-integer or mixed-category optimization. However, such methods cannot handle the three types of variables simultaneously. In this study, we propose CatCMA with Margin (CatCMAwM), a stochastic optimization method for MV-BBO that jointly optimizes continuous, integer, and categorical variables. CatCMAwM is developed by incorporating novel integer handling into CatCMA, a mixed-category black-box optimization method employing a joint distribution of multivariate Gaussian and categorical distributions. The proposed integer handling is carefully designed by reviewing existing integer handling and following the design principles of CatCMA. Furthermore, we extend CatCMAwM to multi-objective MV-BBO by instantiating it within the Sofomore framework, obtaining a multi-objective optimizer termed COMO-CatCMA with Margin (COMO-CatCMAwM). Numerical experiments on single-objective MV-BBO problems show that CatCMAwM effectively handles the three types of variables, outperforming state-of-the-art Bayesian optimization methods and baselines that simply incorporate existing integer handlings into CatCMA. Moreover, on bi-objective MV-BBO benchmarks, COMO-CatCMAwM achieves competitive or superior hypervolume compared to representative baselines.