One Generalization of Graph Isomorphism Problem
I wasn't able to figure out whether $\mathrm{GI}(1/2)\in\mathsf P$, but here is a weaker result: The languages $\mathrm{GI}(1/2)$ and $\mathrm{GI}$ are not $\mathsf P$-inseparable.

Proof: It suffices to show the first reverse inclusion in
$$\mathrm{GI}(1/2)\supseteq\{(G,H): |V_G|=|V_H|, |E_G|=|E_H|\} \supseteq \mathrm{GI}(1)=\mathrm{GI}.$$
Let $p$ be the probability that $(a,b)\in E_G$ for uniformly random $a$, $b$.

Let $\Psi:V_G\to V_H$ be a uniformly random function and also let $a$ and $b$ be selected randomly, with $\Psi$, $a$, $b$ mutually independent.
Then the two events
$$\mathscr U = \{(a,b)\in E_G\}\text{,}\quad \mathscr V=\{(\Psi(a),\Psi(b))\in E_H\}$$
are independent. Therefore
$$\Pr_{\Psi,a,b}(\mathscr U\leftrightarrow\mathscr V)=p^2+(1-p)^2\ge 1/2.$$

So by Fubini's Theorem there is some $\psi$ such that
$\Pr_{a,b}(\mathscr U\leftrightarrow\mathscr V\mid \Psi=\psi)\ge 1/2.\qquad\Box$