In KKT, we have some optimal point $x^*$ with associated $\mu^*$ value for inequality constraints. Are these $\mu^*$ values unique for a given $x^*$ (for the primal problem)?
It seems that when these constraints turn active (you go from inequality to equality constraint) the exact magnitude of $\mu^*$ doesn't actually matter since you are bound by primal feasibility anyways (specifically $g(x) \leq 0$) so you can't improve your score by moving in the direction of the $f(x)$ gradient when it becomes perfectly opposing $g(x)$. Thus, although your solution for a $\mu < \mu*$ on an active constraint no longer satisfies the lagrangian gradient $= 0$ condition but is nontheless a local optimum.
The scale of $\mu$ is bound by stationarity, indeed when you have the derivative of the Lagragian equal to $0$, the term $\mu^T\partial g(x)=b(x)$ has a unique solution with the additional constraint of dual feasibility and complementary slackness making sure that some $\mu_i$ are $0$. The solution is unique since active constraints must have independent normal directions.