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Slice quaternionic analysis in two variables is a generalization of the theory of several complex variables to quaternions. This study relies on the theory of stem functions and the theory of holomorphic functions in two complex variables. Our approach is to introduce holomorphicity for stem functions in terms of two commutative complex structures. It turns out that, locally, a function which is slice regular corresponds exactly to the Taylor series of two ordered quaternions, with the coefficients on the right. The Hartogs phenomenon holds in our setting; however, its proof is subtle due to some topological obstacles. We overcome them by showing that holomorphic stem functions preserve the property of being intrinsic after extension.