Geometrically, why is the same wave equation valid for both transverse and longitudinal waves?

There is actually an interesting geometric explanation for this phenomena, and the logic of it is essentially the same as the case for deriving the transverse wave equation. For detailed derivation with schematic please see page 4 of this document https://scholar.harvard.edu/files/david-morin/files/waves_longitudinal.pdf , I'll do a heuristic explanation.

The wave equation in general is a description of the fact that there is a restorative force acting on the "medium" or "fields", proportional to the curvature of those fields(2nd spatial derivative). Since that force is then proportional to the acceleration of the medium/force (2nd time derivative), we can therefore state that the 2nd time derivative is proportional to the 2nd space derivative.

This comes up with transverse waves when you consider a small piece of rope with tension pulling it at an angle from one side and tension pulling it at an angle from the other side, those angles are the first derivative (because they are tangent to the rope), and since we have to take their difference them, we end up with a "pull" on the rope proportional to the 2nd derivative. And at this point we're done! We've shown that the restorative force due to tension is proportional to the curvature of the string. Boom. Wave equation.

Now how about for longitudinal waves? Let's consider sound traveling through a "tube" or an enclosed medium. Now if we have a segment of that tube, at every point in time there is some sound affecting the boundaries of the segment, the whole segment moves because the sound is "pushing it" forward, but because the "push" is not equal on the boundaries there is going to be uneven pressure and therefore a change in volume in that segment, if you run through the calculations (basic pressure, area, volume stuff), you see that the change in volume of the tube segment is directly proportional to the amount that the the segment has moved forward. Now we are in familiar territory and can apply Newton's second law.

The document I linked goes through all of the calculations, but I hope my explanation helped show that waves are just a manifestation of restorative fields, longitudinal vs. transverse is a meaningless distinction at that level of generality.