Szabolcs, thanks for your reply.
Yes, it seems that Wolfram has made a sound mistake on the very roots of the definition of the Mathaematica Graph tool.
However, I gather from the appreciated, and smart, reply by Jaebum Jung in
http://community.wolfram.com/groups/-/m/t/603839?ppauth=qNrI6umO
that there would be a way out to the issue without need to expect a redefinition of the foundations of the concept.
In that direction, I wonder whether WRI could (or actually should) provide a free patch under the form of a package including a set of routines for the missing properties of the graphs with duplicated edges. After all, it is a latent or hidden vicious of the product, rather than an inadvertent mistake on a lateral or minor thing made by the designers.
For example, along with the available standard set of edges provided by EdgeIndex (which 'ignores' the duplicated edges), a possible solution could be a standard package providing another internal or parallel an different index associated to the positions of the whole edges in the EdgesList). Solving the problem depicted in the image below (part of the notebook I attached in the link above)
There is no distinction between the first "M" \[DirectedEdge] "V" (in 'edgesRed') and the last "M" \[DirectedEdge] "V" (in 'edgesPurple'). They carry the same name because they connect the same pair of vertices but they are in different contexts and are different objects.
(* edge indices of 'MVPQGraph' *)
ei = EdgeIndex[MVPQGraph, #] & /@ EdgeList[MVPQGraph]
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 13, 7, 10, 5}
(* position edge indices *)
pei = Flatten[Position[EdgeList[MVPQGraph], #] & /@ EdgeList[MVPQGraph]]
{1, 2, 3, 11, 4, 12, 5, 16, 6, 7, 14, 8, 9, 10, 15, 3, 11, 4, 12, 13, 7, 14, 10, 15, 5, 16}
The sophisticated solution proposed by Jaebum seems to set the way. I havent explored it in depth yet, but, in any a case it is too tricky for being of any use for the common Mathematica user or even for the advanced: I guess the Mathematica is a tool not an end for most of us.
It is my view in case it be of any use for the Community.
Martin.