Group isomorphic to $\Bbb Z/3\Bbb Z\times \Bbb Z/3\Bbb Z$ Is a group with $9$ elements such that all elements (excepted the natural element) are of order $3$ is isomorphic to $\Bbb Z/3\Bbb Z\times \Bbb Z/3\Bbb Z$ ?

Yes: since a group of order $p^2$ ($p$ prime) is abelian, we can use the structure theorem for finite abelian groups. A group of order $9$ is either cyclic, isomorphic to $\mathbf Z/9\mathbf Z$ or isomorphic to $\mathbf Z/3\mathbf Z\times\mathbf Z/3\mathbf Z $ .