The interpolation step in the Guruswami-Sudan algorithm has attracted a lot of interest and it is now solved by many algorithms in the literature; they use either structured linear algebra or basis reduction for polynomial lattices. This problem of interpolation with multiplicities has been generalized to multivariate polynomials; in this context, to our knowledge only the approach based on polynomial lattices has been studied until now. Here, we reduce this interpolation problem to a problem of simultaneous polynomial approximations, which we solve using fast algorithms for structured linear algebra. This improves the best known complexity bounds for the interpolation step of the list-decoding of Reed-Solomon codes, Parvaresh-Vardy codes or folded Reed-Solomon codes. In the special case of Reed-Solomon codes, our approach has complexity $\mathcal{O}\tilde{~}(\ell^{\omega-1}m^2n)$, where $\ell,m,n$ are the list size, the multiplicity and the number of sample points, and $\omega$ is the exponent of linear algebra; the speedup factor with comparison to the state of the art (Cohn and Heninger's algorithm) is $\ell / m$, with $m \leqslant \ell$ in this context.