As far as I can tell, I would have to answer that in the negative (see the EDIT in the previous post). In general, the set of graphs is more complicated/general than the set of countable/finite lattices (every lattice can be represented as a DAG by setting a→b iff a∧b=a and b→a iff b∨a=a), which leads me to believe that there is no obvious isomorphism between operations.

There may be some weirdness where we can take subsets of vertices/edges of graphs and then endow those subsets with lattice structure, which is somehow isomorphic to the original graph, but this is again, not obvious to me how it would be done.