This is a bad metaphor for rubes, as you say. The axle gives a conserved quantity across the transition for a wagon, the distance between the wheels is constant as the transition is happening. So the bending goes to 90 degrees when the boundary becomes parallel to the direction of cart propagation. This is not what happens with light, light refracts at a finite angle less than 90 degrees for light coming in nearly parallel to the surface, this is the angle past which you have total internal reflection in the medium.
The reason is that the perpendicular horizontal distance along the wavefronts is not the quantity that is conserved when light enters a stationary medium, like it would be if photons had little axles (they don't). The quantity that is conserved across the refraction transition is the frequency of the light, the energy of the photons. The reason is that the material provides a time-independent propagation environment, and in a time-independent background, the energy is conserved. Classically, the modes for a time-independent medium are found by separation of variables with a fixed frequency in time, and this is saying the same thing but without using quantum mechanics to relate conservation of frequency to conservation of energy.
So the analog of the axle length in this case is the time between wave-crests crossing a given point. This means that the wavelength in the medium is reduced by the index of refraction (to keep the frequency of crests crossing a given point constant), so that the outside wavelength is $\lambda$ and the interior wavelength is $\lambda/n$.
If medium surface lies parallel to the x-axis, and the incoming light wave crests make an angle of $\theta$ with respect to the x-axis ($\theta=0$ is crests parallel to the x-axis, so the light is coming head on, and no refraction), then the distance between the points where successive wave crests hit the medium boundary is $\lambda/\sin(\theta)$. In the interior of the medium, the same argument tells you that it's $\lambda/n\sin(\alpha)$ where $\alpha$ is the angle of the crests with respect to the x-axis in the medium, so in order for the crests inside to match the crests outside, you need
$$ \sin(\theta) = n \sin(\lambda)$$
and this is Snell's law for the case where the n outside is 1. You can always consider the wave-speed outside to be 1, so the quantity n is the ratio of the speed of waves inside to outside, and so it isn't really a special case. Also note that the same law holds for sound refraction, or any wave.