The definitions I came across all assumed that the adjacency matrices are unweighted. (Look at e.g. the definition for the conormal/disjunctive product, which explicitly assumes 0-1 matrices.) There may be a way to modify these to work consistently for weighted matrices, but I am not aware of how to do so.

But consistency is already there, WeightedAdjacencyMatrix is the same as AdjacencyMatrix for not weighted graphs, so seems like replacing AdjacencyMatrix with WeightedAdjacencyMatrix and AdjacencyGraph with WeightedAdjacencyGraph (after replacing 0 with Infinity, which I find to be weird inconsistency between two functions) is a natural generalization for all the products.