Linearity implies the superposition principle. The superposition principle means that if $\psi_1(x,t)$ is a solution to the (vacuum) wave equation, and $\psi_2(x,t)$ is also another solution to the same equation, then $\psi=\psi_1+\psi_2$ will also be a solution. Interaction in the most general (non-QFT) sense would require that a wave configuration $\psi_1(x)$ "senses" the presence of another wave configuration $\psi_2(x)$, and gets modified by it. But the assumption that $\psi_1(x)$ is modified by $\psi_2(x)$ means that its time evolution is different compared to the situation that $\psi_2$ is not present. Hence, the time evolution of two wave configurations cannot be just a superposition which is ignorant of whether the respective other configuration is present or not.
If linearity precludes interaction in the above sense, then non-linearity is at least a necessary condition for interaction. It is not generally also a sufficient condition because any linear equation system can be made apparently nonlinear by a sufficiently complicated coordinate transform. And yet, the physics behind it cannot be changed by using different coordinates.
Light by light scattering is not a proof that light is not "wave-only". You can also include non-linear material laws into electrodynamics, which are able to describe certainnumerous macroscopic phenomena, like you can look up in the nonlinear Stark effectWikipedia article on nonlinear optics, which also cause light-by-light scattering (in matter). The point about being not "wave-only" is whether apparent macroscopic laws look more granular if you look closer. Just like the Navier-Stokes equations of hydrodynamics are only valid as long as you don't ever look with a microscope at the Brownian motion of dust particles suspended in the fluid, the Maxwell equations in matter are only valid macroscopically.