Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$? On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.  
Question:  Does $g$ satisfy uniqueness up to homothety in the following sense: if $h$ is another metric satisfying $\mathrm{Ricc}(h)=-g$ then $h=cg$ for some $c>0$?
 A: so here I guess $M$ is a compact Kähler manifold. 
Thanks to Yau theorem, we know that there exists a unique Kähler metric $h$ in each Kähler cohomology class such that $\mathrm{Ric}(h)=-g$ (more generally you can prescribed the Ricci form to be any form in the cohomology class of $c_1(M)$).
So to answer your question, there are a lot of such metrics $h$, but there is only one such in each Kähler cohomology class of $M$.
In particular, if $h$ lies in a (positive) multiple of $-c_1(M)$, then $h$ is proportional to $g$. 
A: In general with your assumption
$$Ric(\omega)=\sqrt{-1}\partial_i\bar\partial_j\log\det g_{i\bar j}$$
so by this formula we have always $Ric(cg)=Ric(g)$, and if $h,h'$ satisfies in $Ric(h)=−g$, $Ric
c(h')=−g$ then in general it is sufficient to use of this formula and remove $\partial\bar\partial$ of bothsides and we get $\omega^n=e^c\omega'^n$ or $\det h_{i\bar j}=e^c\det h'_{i\bar j}$ for a constant $c$, so a lot of such metrics exists
and by Aubin's proof (he was a first French Mathematician who proved it and later Yau gave different proof) and Calabi himself also gave a proof for unicity of such metrics when the first Chern class is negative
if the kahler metric $h'$ lies in a $ c_1(K_X)$,then $h=h'$. See Tian's book 
In fact if $\omega'$ be the corresponding metric of $h'$ then by $dd^c$-lemma we can write $\omega'=\omega+\sqrt{-1}\partial\bar\partial \varphi$ and so we will have a non-linear parabolic monge-Ampere equation 
$$\frac {(\omega+\sqrt{-1}\partial\bar\partial \varphi)^n}{\omega^n}=e^c$$
which its solution is unique. See theorem 19.1 page 129 Lectures on Kahler geometry  Andrei Moroianu
