Suppose $G$ is a group. Lets consider its trivial subgroup, i.e. $\{e_G\}$. Since trivial subgroup is normal in $G$ then $G/\{e_G\}$ is group. From the definition of quotient group we know that $$G/\{e_G\}:=\{g\{e_G\}:g\in G\}$$ since $g\{e_G\}=\{ge_G\}=\{g\}$ then $G/\{e_G\}=\{\{{g}\}:g\in G\}$. Could we say that $G/\{e_G\}=G$ i.e. they are the same. I know that they are isomorphic but is it correct to say that they are identical.
If not please explain that.
No, you cannot say $G/\{e_G\}=G$.
If $g$ is an element of $G$, then $G/\{e_G\}$ contains the element $\{g\}$, which is not the same. They are, however, clearly isomorphic.