In interactive proofs, do we assume that the prover can always solve the problem?
No, but if the problem isn't known to be in promiseMA then

we do "assume that the prover can always solve the problem".

https://cstheory.stackexchange.com/questions/696/mip-with-efficient-provers

Graph isomorphism is trivially in NP, so for graph isomorphism, we just

"assume that the prover" has an isomorphism between the graphs.

Graph non-isomorphism is not even known to be in MA, so for graph non-isomorphism,

we do "assume that the prover can always solve the problem".