Decoupling multivariate polynomials is useful for obtaining an insight into the workings of a nonlinear mapping, performing parameter reduction, or approximating nonlinear functions. Several different tensor-based approaches have been proposed independently for this task, involving different tensor representations of the functions, and ultimately leading to a canonical polyadic decomposition. We first show that the involved tensors are related by a linear transformation, and that their CP decompositions and uniqueness properties are closely related. This connection provides a way to better assess which of the methods should be favored in certain problem settings, and may be a starting point to unify the approaches. Second, we show that taking into account the previously ignored intrinsic structure in the tensor decompositions improves the uniqueness properties of the decompositions and thus enlarges the applicability range of the methods.