Before to make my question, I'm going to introduce all necessary notions.
Let $G$ be a linearly reductive group (i.e. each rational representation is completely reducible) acting regularly on an affine variety $X$. Given a (rational) character of $G$, $\chi : G \rightarrow \mathbb{C}^\times$, we have an associated action of $G$ over $X \times \mathbb{C}$ by $g \cdot (x,z)=(g\cdot x,\ \chi(g)^{-1}z)$. At this point we can define the associated Proj quotient of X by G (which is well defined) by: $$X//_\chi G := Proj\bigl(\mathbb{C}[X \times \mathbb{C}]^{G,\chi}\bigr)$$ where the gradation is given by $\mathbb{C}[X]^{\chi^n}=\bigl\{f \in \mathbb{C}[X] \ : \ f(g\cdot x)=\chi(g)^nf(x) \bigr\}$. This scheme is in particular a quasi-projective variety.
At the same time we can define the notion of $\chi$-semistable points in $X$ as $x \in X$ such that $\exists f \in \mathbb{C}[X]^{\chi^n} \ s.t. \ f(x)\neq0$. Therefore, the locus of $\chi$-semistable points $X^{\chi}_{ss}$ is an open $G-stable$ subset in $X$. At this point we can take the quotient of $X^{\chi}_{ss}$ by the closure equivalence relation (i.e. two points are equivalent if the closures in $X^{\chi}_{ss}$ of their orbits have non-empty intersection) $X^{\chi}_{ss}/\sim$.
Then, we have an homeomorphism $X//_\chi G \cong X^{\chi}_{ss}/\sim$.
My question: If $\chi'$ and $\chi''$ are two $G-$character inducing the same semistability notion on $X$, then we have an homeomorphism $X^{\chi'}_{ss}/\sim = X^{\chi}_{ss}/\sim$ and therefore an homeomorphism $X//_{\chi''}G = X//_{\chi'}G$. Is the last an isomorphism as quasi-projective varieties/scheme?