In recent times, the mixed multifractal analysis has generated interest. The purpose of this paper is twofold. Firstly, we define the mixed multifractal Hewitt-Stromberg measures and introduce a multifractal formalism based on this type of measures. Secondly, we prove that the upper mixed multifractal box-dimension of the overlaps of self-similar sets and measures is strictly smaller than the mixed multifractal Hausdorff dimension of self-similar sets and measures. Finally, we demonstrate that this result is still valid for the mixed multifractal Hewitt-Stromberg dimensions.