I have this question in my mind since a long time. But I am failed to find an answer.
The following figures are closed, connected, orientable surfaces of genus $2$.
Topologically both surfaces are same. Moreover, their first fundamental group is same, i.e., $$ \pi_1 (S) = \left\langle a_{1}, b_{1}, a_{2}, b_{2} \bigg| [a_{1}, b_{1}] [a_{2}, b_{2}] \right\rangle$$
Is the same true if we see this via Teichmuller theory?
We define representation variety of the fundamental group $\pi_1(S)$ of a closed connected surface $S$ of genus $g>2$ , with values in a Lie group $G$ as follows: $$\text{Rep} \big(\pi_{1} (S), G\big) := \hom \big(\pi_{1}(S), G\big) / G$$ where $G$ acts on $\hom (\pi_{1}(S), G)$ by conjugation. Now, Teichmuller space is a connected component of the representation variety $\hom(\pi_1(S), PSL(2, \mathbb{R}))/PSL(2, \mathbb{R})$. Let denote the first and second surfaces by $S_1$ and $S_2$ respectively. My question is are both $\text{Teich}(S_1)$ and $\text{Teich}(S_2)$ same?
Yes, when $S_1, S_2$ are closed connected oriented surfaces of the same genus, then their Teichmüller spaces are homeomorphic. They are both homeomorphic to $\mathbb{R}^{6g-6}$. The way you define Teichmüller spaces as connected component of a space of representation, they are also homeomorphic, since the two fundamental groups are isomorphic and thus you get the same representations.
Literally as a set, the Teichmüller spaces might not be the same, just like $S_1$ and $S_2$ are not the same as sets, only as topological spaces.