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I would say that if you split all edges of both graph by adding adding a new vertex b joined to a and c where there was an edge from a to c (this is a way to transform a graph with multiple edges into a normal graph), then checking that one is a minor of the other would give you the good definition.

I expect it to be slower, though.

And the explanation to your question "why aren't multiple edges considered by the minor function" is very easy and natural : why should it ? Things get implemented if somebody who cares about them takes the time to do it. Otherwise, they are not implemented.