Determine whether the following graphs are isomorphic.
Labelling vertices of both graphs as $u_1,u_2,u_3,u_4,v_1,v_2,v_3,v_4$ in the order given above, we see that these graphs are bipartite with $u_i\leftrightarrow v_j$ iff $i\ne j$. Is showing this correspondence enough to prove that they are isomorphic? Is that because the labellings allow us to write adjacency matrices, equivalency of which supports isomorphism between them?
A graph isomorphism $G_1\cong G_2$ is by definition a bijection $\varphi\colon V(G_1)\to V(G_2)$ between the sets of vertices of $G_1$ and $G_2$ such that $(u,v)$ is an edge in $G_1$ if and only if $(\varphi(u),\varphi(v))$ is an edge in $G_2$.
Your labeling of vertices achieves exactly that: you have constructed a bijection between the sets of vertices and verified that it send edges to edges and non-edges to non-edges.
Hence, you have shown $G_1\cong G_2$.