We study the sets of points where a L\'evy function and a translated L\'evy function share a given couple of H\"older exponents. The multivariate multifractal spectrum, i.e., the function which, for each couple, associates the Hausdorff dimension of the corresponding set, is estimated and we study its dependency on the translation parameter. On the other hand, we determine the multivariate multifractal Legendre spectra of shifted L\'evy functions. This allows to explore how the validity of the multivariate multifractal formalism depends on the shift parameter.