In this case, "axiom" is a bit misleading. It's an assumption, but not an axiom.

Relativity arose out of the observation that the speed of light was the same for all observers, which had become clear by the time relativity was developed. What relativity does is goes back and says "okay, what assumptions about physics are wrong in order for that to be possible?"

It turns out the wrong assumption was the idea that all observers, regardless of position or movement, agree on lengths in space and time.