There is no special crime "assault with a deadly weapon" in Washington state, but "deadly weapon" figures into the definition of 1st and 2nd degree assault. Third degree assault refers (as a possibility) to "weapon", and the relevant potentially applicable clause is in terms of an act that
With criminal negligence, causes bodily harm to another person by
means of a weapon or other instrument or thing likely to produce
bodily harm
Deadly weapon has a statutory definition, as
any explosive or loaded or unloaded firearm, and shall include any
other weapon, device, instrument, article, or substance, including a
"vehicle" as defined in this section, which, under the circumstances
in which it is used, attempted to be used, or threatened to be used,
is readily capable of causing death or substantial bodily harm
which doesn't apply to a dog bite. You would have to load the dog with explosives in Washington for it to constitute a "deadly weapon". The definitions do not extend to simple "weapon", so the "ordinary" meaning of weapon would be applied (in the case of 3rd degree assault, where the thing is just "a weapon"). RCW 9.94A.825 stipulates a special verdict associated with deadly weapons, and a definition:
For purposes of this section, a deadly weapon is an implement or
instrument which has the capacity to inflict death and from the manner
in which it is used, is likely to produce or may easily and readily
produce death. The following instruments are included in the term
deadly weapon: Blackjack, sling shot, billy, sand club, sandbag, metal
knuckles, any dirk, dagger, pistol, revolver, or any other firearm,
any knife having a blade longer than three inches, any razor with an
unguarded blade, any metal pipe or bar used or intended to be used as
a club, any explosive, and any weapon containing poisonous or
injurious gas.
(Note that Title 9A is different from title 9 so we get two different versions of "deadly weapon). A dog is not a thing described by this definition.