In a field why does the multiplicative identity have an additive inverse, whereas the additive identity doesn't have a multiplicative inverse?

As I said in the comments, the symmetry is broken because of the distributivity, $a(b+c) = ab + ac$, which does not hold symmetrically (indeed $a+bc = (a+b)(a+c)$ is not true in general).

This identity, the group axioms for $+$ and the fact that $0\neq 1$ altogether imply that $0$ has no multiplicative inverse ($0a = (0+0)a = 0a + 0a$ and so $0a=0$, so unless $0=1$, $0$ has no inverse)

These axioms are there to generalize $\mathbb{Z},\mathbb{Q}$ (integral domains, or more generally rings with more than one element), and are therefore "natural", because in particular distributivity is essential in those structures.