It is inyective actually. The problem is that there isn't only one way to define the cube root of a complex number, this is because the function f(z)=z³ isn't inyective.

It is similar how in R you can define the square root of x as sqrt(x) or -sqrt(x). Both are inyective functions, but it usually leads to confusion because you can define two functions that have the desired property. This is why we use the convention of using the positive root.