So I'm still not sure I totally get the objection -- please help me out here. In my mind, multiple hierarchies are represented by a single graph.
When one reasons about the system, one will naturally only consider single hierarchies at a time for simplicity. The way to do this is center yourself on a node at a time, and then from that vantage point, traverse the node in different directions.
In the system, each edge belongs to only 3 categories: P ("parent", directed), A ("associate", undirected), S ("source", directed). We're only considering hierarchies, so we'll restrict ourselves to "P". Each node can have an arbitrary number of relationships with other nodes. A family and work relationship could look like this.
You -P-> your kid 1
You -P-> your kid 2
Mother -P-> You, Sister
Father -P-> You, Sister
Boss -P-> You
Boss -P-> Colleague
Company President -P-> Boss
Company -P-> Company President
Seems to me it should be easy to reason about multiple parents in different contexts -- they all belong in the same graph.
Wikipedia sort of does this with its Categories box at the bottom of every page, but Wikipedia only has one kind of relationship category: P.
I feel like this is missing something, namely in modelling relationships where a single child can have an arbitrary number of parents in a single context (or source-target without cycles.) I don't think that's adequately captured by multiple-hierarchies when you can't break the parents into discrete categories (e.g. mothers and fathers) and thereby into discrete hierarchies. If you try to model a DAG like that as multiple hierarchies by somehow imposing an order onto the set of parents, you'll get yourself into a quagmire.