Errr... Vassal, liege. Two rather different things.
1) Liege DOW somebody, but haven't mobilized all vassals. A vassal then mobilize by themselves and DOW the same target. This is the thing that is mentioned in the FAQ -> sieges initiated by the "separate" vassal can not be "stolen" by the liege.
2) When a vassal DOW somebody the liege gets a call to take part in the war too. It doens't matter if you are the liege or the vassal. If the liege answers the call, then he is in the same war as his vassal -> sieges can be "stolen".