See the
second postulate. A postulate (axiom) is a claim that cannot be proven, but is taken as factually true and as a starting point of logical deductions.
As to the special role given to light, I might comment the following: A more modern approach postulates the existence of a limit speed of propagation of interactions in Nature. In Classical Mechanics, the interaction is described via potential energy that is a function of the instantaneous relative positions of the particles. This is known as "action at a distance", which implies infinite speed of propagation of the interactions.
The existence of a limit speed makes it a fundamental constant of Nature. According to the first postulate of Relativitiy (so called Principle of Relativity), this constant must have the same meaning and value as measured by all inertial observers.
If one develops Electrodynamics according to Hamilton's Principle of least action, one finds that electromagnetic fields can propagate through empty space with the same limit speed. According to Maxwell's theory, light is just an electromagnetic wave. Thus, this limit speed coincides with the speed of light.
But, other fields also propagate with the same speed. For example, according to General Relativity, a disturbance in the metric of flat spacetime also propagates as a gravitational wave with the speed of light.
It also turns out that massless particles necessarily travel at the same speed.