First, as correctly observed, waterwaves and soundwaves are both mechanical waves. The closest analogue to waterwaves are surface acoustic (sound) waves.
Sound waves in a solid may have both a transverse and a longitudinal component. For example, surface acoustic waves and Rayleigh waves have transverse components propagating on the surface of a solid.
However, breaking waves ("curl brake") occur only in surface waves in fluids, and not in a solid. This is because, when the shear strain is large enough, the solid would just break down into pieces, instead of realizing a breaking wave.
In a nutshell, surface mechanical waves in fluids can exhibit breaking waves ("curl brake"). Surface mechanical waves in solids, at certain regimes, may just break the solid into pieces instead.
Both breaking waves and broken solids are nonlinear effects, of course.
Sound waves in air may have shear transverse waves (even though they are kinda exotic). I don't see any physical reason why these shear waves could not break. However, I think breaking waves of these kind have never been observer. This looks like an open research question.
To cite the paper above:
The dramatic process of wave breaking cannot be observed in solids with SAWs, since solids usually break under strain as soon as a certain degree of bond extension (typically 10–20%) has been reached.
Shear (transverse) waves may occur also in air.
Honestly, I think that whether shear waves in air may have breaking waves is an open question.