This is a good question and a nice way to study some complexity-theorems from the '90's. Reviewing Shamir's famous paper, it is not so much "if a particular position is a winning one for white, a machine can prove me this in polynomial time," it is more "if a particular position is a winning one for white, I can engage with a powerful prover machine in a number of rounds of challenge/response to be convinced by the prover that there is a winning position." In the challenge/response, the prover is pretty abstracted from chess and is merely proving that some polynomial over some finite field is non-zero.
For example, $\mathcal{PSPACE}$ refers to problems running in space polynomial to the input size. Much as $\mathsf{3CNF}$ is the standard example of an $\mathcal{NPC}$ decision problem, True Quantified Boolean Formula, $\mathsf{TQBF}$, is the standard example of a decision problem complete for $\mathcal{PSPACE}$. A $\mathsf{QBF}$ formula might be $\forall x\ \exists y\ \exists z\ ((x \lor z) \land y)$. The relation of $\mathsf{TQBF}$ to games such as chess or checkers is noted, in that a "white-to-mate-in-$n$" problem can also be cast as
$$\exists w_1\forall b_1\exists w_2\forall b_2\cdots\exists w_n:\phi(w_1,b_1,w_2,b_2,\cdots,w_n)$$
where $\phi$ is a Boolean encoding the rules of chess as applied to the given position. We can read this as "there is a move by white, such that for all moves by black, there is a counter-move by white, such that... white checkmates black.
Turning to $\mathcal{IP}$, interactive proofs involve two parties, say $P$ the prover and $V$ the verifier. $V$ and $P$ engage in rounds of challenges and responses. The "interactive proof" of graph non-isomorphism $\mathsf{GNI}$ given in the question is typically framed as a single round of challenge/response, that is amplified to improve soundness, say, 100 times. However, $\mathcal{IP}$ as used by Shamir involves a polynomial number of rounds of interaction between $V$ and $P$.
The denouement, of course, is that $\mathcal{IP}=\mathcal{PSPACE}$. Shamir and his predecessors showed (1) how to convert any $\mathsf{TQBF}$ problem into a canonical form ("simple QBF"), (2) how to "arithmetize" the simple QBF by replacing each $\forall$ with $\Pi$, each $\exists$ with $\Sigma$, each $\land$ with multiplication, etc. thus converting a simple QBF to a polynomial, and (3) how to have a verifier, $V$, challenge a prover, $P$, with random interactive coin tosses to establish that the polynomial is nonzero. Each round of the interaction simplifies the polynomial, and eventually, as long as $V$ is randomly choosing her coin tosses for each round after $P$ provides her answers, $V$ should be convinced by $P$,
However, $P$ really has to work hard to answer $V$'s challenges. For example, she must actually evaluate very large polynomials for each round. The work that the prover has to do does not seem very enlightening with respect to chess itself.
