Let $G,H$ be graphs on vertices $V(G)$ and $V(H)$ respectively. Let $f:V(G)\to V(H)$ be a bijection which preserves vertex degree i.e $$\forall u\in V(G)\left (\deg _Gu=\deg _Hf(u)\right ) $$ Is this sufficient for $G\cong H$?
Since degrees are preserved, it must mean there are equally many edges, since the sum of degrees is exactly twice the number of edges.
Does the same $f$ fit the bill, then? Is $\{u,v\}\in E(G)$ equivalent to $\{f(u),f(v)\}\in E(H)$? NOPE!!
No. Consider the cycle graph $C_6$, and the disjoint union of two cycle graphs $C_3 \sqcup C_3$. Both have six vertices of degree 2, but they are not isomorphic.