We report the thermodynamic properties, magnetic ground state and microscopic magnetic model of the spin-1 frustrated antiferromagnet Li2NiW2O8, showing successive transitions at TN1≃18 K and TN2≃12.5 K in zero field. Nuclear magnetic resonance and neutron diffraction reveal collinear and commensurate magnetic order with the propagation vector k=(12,0,12) below TN2. The ordered moment of 1.8$\mu$B at 1.5 K is directed along [0.89(9),-0.10(5),-0.49(6)] and matches the magnetic easy axis of spin-1Ni2+ ions, which is determined by the scissor-like distortion of the NiO6 octahedra. Incommensurate magnetic order, presumably of spin-density-wave type, is observed in the region between TN2 and TN1. Density-functional band-structure calculations put forward a three-dimensional spin lattice with spin-1 chains running along the [011] direction and stacked on a spatially anisotropic triangular lattice in the ab plane. We show that the collinear magnetic order in Li2NiW2O8 is incompatible with the triangular lattice geometry and thus driven by a pronounced easy-axis single-ion anisotropy of Ni2+.