These are two techniques I think about quite a bit and was wondering if there is a meaningful connection.
(Both techniques are relevant to a problem I'm working on, although they apply to different aspects.)
Disjunctive sum is an operation that takes games and produces a game where players move in exactly one component at a time. If you know the nimbers/grundy values of the components, then the nimber/grundy value of the disjunctive sum is given by taking the values for the components and adding in binary without carrying.
But the "combining" you do when combining partial products for binary multiplication is adding with carrying.
Also, disjunctive sum is much more general than just the "add in binary without carrying operation" that is useful when you're taking the disjunctive sum of impartial games and want information about 2-player normal play and can effectively compute the nimbers/grundy values of the components.