Quote Originally Posted by Didz
I'm not sure if this is meant to imply that they are client kingdoms of each other, or one of the other, or that they are both client kingdoms of someone else

The faction overview shows the HRE as a vassal of Denmark, but makes no mention of France as a vassal of anyone. Not only that but the HRE seems to have its own independant set of allies (Milan, Byzantine, Russia, Denmark, and Hungary) which is quite different to Denmarks (England, Venice and Poland)
It should mean that France is a vassal to the HRE. Vassals still can have their own allies and enemies.

On a secondary point: Can anyone confirm what impact vassalage has on Faction Goals and diplomacy, for example, if Scotland were to make England a vassal state would that satisfy their faction goal for the Short Campaign or does elimination mean exactly that, also is is possible to declare war on a vassal without going to war with its protector?
I´ve never tried to make the faction I had to eliminate my vassal, so I can´t answer the first part. The second part, however, is simple: Yes, you can declare war on a faction that´s a vassal to someone else without automatically triggering a war with the overlord. However, the overlord will not like it very much that you attacked his vassal (the same goes for allies, by the way) so expect a drop in your relationship. And if you´re allied with the overlord you´ll most likely lose your alliance, unless you can somehow get the vassal to initiate the war.