The limit of speed of light is a property of space-time, not a property of light.
Another way to understand it is that it is the maximum speed of causality.
The reason only light can achieve this speed is that it is massless. Because if light had mass, it would have infinite energy, and take infinite energy to achieve the speed of light.
There are numerous explanations for how 2 objects moving at the speed of light still only approach each other at the speed of light.
Take a look at Youtube, there are many good videos explaining relativity in general and speed of light in particular.