Firstly, I just want to make sure that I've understood the notions of relative and absolute quantities correctly.
Elementary analysis shows that position and velocity are relative quantities. Indeed, position is clearly relative as two inertial frames $S$ and $S'$ displaced by a constant displacement vector $\mathbb{r}_{0}$ will measure the position of an object to be at $\mathbb{r}$ and $\mathbb{r}'$ respectively, the two positions related by $\mathbb{r}=\mathbb{r}'+\mathbb{r}_{0}$. As these two frames are arbitrary and neither can be distinguished from the other as a preferred absolute rest frame, it must be that position is relative. This argument also holds if the two frames $S$ and $S'$ are in relative motion to one another, related by $\mathbb{r}=\mathbb{r}'+\mathbb{v}t$, where $\mathbb{v}$ is the relative velocity between the two frames. Clearly it follows from this (by differentiating with respect to time) that velocity is also relative.
Now, if I understand it correctly, Newton introduced the notion of absolute space, and thus defining the absolute position and velocity of a given object as their position and velocity measured relative to this frame. Thus these relative quantities defined in the previous paragraph are all related to absolute quantities (that in principal will be the same for all observers at rest relative to absolute space, regardless of where they are located in this space). However, as a result of Galileo's principal of relativity ruling out the existence of a frame at absolute rest, i.e. absence of absolute space, it follows that the concepts of absolute position and velocity do not exist and therefore are truly relative quantities, dependent on the frame that they are measured in.
Secondly, if we consider Maxwell's equations, which are not invariant under Galilean transformations, but we require them to hold in all inertial frames, doesn't it immediately follow that the speed of light has the same constant value in all inertial frames from this assumption (given that Maxwell's equations imply a constant speed of light). Why is it given as an axiom of special relativity?