This is Lemma 2.2.5 from Katok's Fuchsian Groups:
Let $\Gamma$ be a subgroup of $\text{PSL}(2,\mathbb{R})$ acting properly discontinuously on $\mathcal{H}$, and $p \in \mathcal{H}$ be fixed by some element of $\Gamma$. Then there is a neighborhood $W$ of $p$ such that no other point of $W$ is fixed by an element of $\Gamma$ other than the identity.
The proof is given below:
Why did we assume $\text{T}$ is not identity? I think this assumption was never used while proving the lemma. Am I missing a crucial point in this proof?
You haven't missed anything. The lemma proves that fixed points of properly discontinuous actions of isometries of $\mathbb{H}$ are isolated. If one takes $T \in \text{PSL}(2,\mathbb{R})$ to be the identity, all points $p \in \mathbb{H}$ are fixed and the argument above is run for an arbitrary $p \in \mathbb{H}^2$. Inspecting the argument in this setting, the above proof shows that if $\Gamma < \text{PSL}(2,\mathbb{R})$ acts properly discontinuously on $\mathbb{H}$, there is a neighborhood of $p$ containing no fixed points of non-identity elements of $\Gamma$, except possibly $p$. This is true, and the statement is in fact seen to be equivalent to the lemma.