We investigate the hydrodynamic behavior and local equilibrium of the multilane exclusion process, whose invariant measures were studied in our previous paper \cite{mlt1a}. The dynamics on each lane follows a hyperbolic time scaling, whereas the interlane dynamics has an arbitrary time scaling. We prove the following: (i) the hydrodynamic behavior of the global density (i.e. summed over all lanes) is governed by a scalar conservation law; (ii) the latter, as well as the limit of individual lanes, is the relaxation limit of a weakly coupled hyperbolic system of balance laws that approximates the particle system. For the hydrodynamic limit, to highlight new phenomena arising in our model, a precise computation of the flux function, with the transitions between different possible shapes (and a physical interpretation thereof), is given for the two-lane model.