Is there a characterization result/some sufficient conditions that ensure that a graph has a unique max flow?
Note that it does not say anything about the min-cuts: a path with all edges having weight 1 has a unique max-flow, but many min cuts.
I believe that it is sufficient that in all optimal solutions of the maximum flow we have that all edges have some positive flow through them. Is that the case?
Counterexample to your conjecture: