It’s the weak point of the analogy, surface gravity waves like you’d get in a shallow lake do have nonlinear behaviour though.

Maybe a more accurate description would be to describe the wavespeed of the medium having tiny variations that cause extremely small, close range kinks where the wavefront crosses past itself, relating speed through time vs speed through space as the radial and tangential propagation of the wavefront. But that’s a less clean analogy, and the lake ripple is still good for describing how an entire universe can appear to be in the middle no matter where you look, despite originating from (suspectedly) a singular point.