We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate-parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of species with quadratic porous-medium interactions in a bounded domain $\Omega$ in any spatial dimension. The cross interactions between different species are scaled by a parameter , with the $\delta =0$ case corresponding to no interactions across species. A smallness condition on $\delta$ ensures existence of solutions up to an arbitrary time $T>0$ in a subspace of $L2(0,T;H1(\Omega ))$. This is shown via a Schauder fixed point argument for a regularised system combined with a vanishing diffusivity approach. The behaviour of solutions for extreme values of $\delta$ is studied numerically.