Given that $ab=e$, we know $b =a^{-1}$. Since $G = \langle a,b\rangle$ this implies $\langle a,b\rangle=\langle a,a^{-1}\rangle=\langle a\rangle=G$ (as elements that generate the group). The presentation rewritten in terms of $a$ is trivial, i.e. $\langle a,a^{-1}\mid aa^{-1}=e\rangle$, so $G$ is a free group on the generator $a$, which is isomorphic to $\mathbb{Z}$.
I have just learned about group presentations and I am also wondering if there is a general method for identifying the isomorphism classes of a group given its presentation.