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A common problem in clinical trials is to test whether the effect of an explanatory variable on a response of interest is similar between two groups, e.g. patient or treatment groups. In this regard, similarity is defined as equivalence up to a pre-specified threshold that denotes an acceptable deviation between the two groups. This issue is typically tackled by assessing if the explanatory variable's effect on the response is similar. This assessment is based on, for example, confidence intervals of differences or a suitable distance between two parametric regression models. Typically, these approaches build on the assumption of a univariate continuous or binary outcome variable. However, multivariate outcomes, especially beyond the case of bivariate binary response, remain underexplored. This paper introduces an approach based on a generalised joint regression framework exploiting the Gaussian copula. Compared to existing methods, our approach accommodates various outcome variable scales, such as continuous, binary, categorical, and ordinal, including mixed outcomes in multi-dimensional spaces. We demonstrate the validity of this approach through a simulation study and an efficacy-toxicity case study, hence highlighting its practical relevance.