Special Properties Of DAWGs with Unique Elements
I could be missing something, but if you took an arbitrary DAG, assigned unique IDs to each vertex, and constructed all words formed by traversing monotone chains from top to bottom, then wouldn't the DAWG of that set be the original DAG ? If so, then a DAWG has no special structure. 

Update: Peter Taylor's comment is the key difference. only one node with indegree 0 unlike a regular DAG.