> As for how that applies to a hash function, I think the lookup gives some sort of weighting for each possible output, for each input, with the largest weightings corresponding to a "match"

Note that in general one "matches" more than one thing in which case (IIRC) the result will be a weighted linear combination of the query results.

I do think that this collapses to a datastructure equivalent in function to the "normal hash table" (but not in performance, lol) in the degenerate case where every input yields strictly a single result with zero activation in all other results, so it's not invalid to call it a differentiable lookup table.