c is a constant of the universe, it's not related to light :)
There are some important constants of the universe, one of them is c. Let's call it "c" without a friendly name in English.
You can (theoretically) measure c in the laboratory without in pure mechanical experiments in experiments that are not about light. The easiest experiment that I can imagine now is the change of the half life of the muons https://en.wikipedia.org/wiki/Time_dilation_of_moving_partic... but I'm not sure to classify it as "pure mechanical experiment".
c is the only speed that doesn't change when you go from a inertial reference frame to another inertial reference frame (i.e. if you have a train moving at a constant velocity and you measure something inside the train or measure it though the window from the platform.)
For some technical reasons the speed of all massless particles must be c. (Because they must travel at the same velocity in every reference frame, because all the reference frames are equivalent.)
And all the waves that can travel in empty space for a long distance must be massless. So the photons and gravitons travel at c. (The theory of gravitons is far for complete, but just ignore this detail.)
For historical reasons, the first time that someone measure c was while trying to measure the speed of light. So they called it "the speed of light". Much later Physicist realized that c was an important constant of the universe and not only the speed of light, but the name stuck and is too late to change it.
So ...
* gravitational waves travel at c
* light waves travel at c. Actually it's better to define mu0=1/(c^2.epsilon0) or something like that. In special relativity electric and magnetic fields are very related, (it's similar to the relation between space and time, but not exactly the same relation). So it's "better" to think that they have only one constant and c, than thinking that they have two independent different constants. Physicist still use two constants, for historical reasons and because sometimes it makes the calculation easier.