I agree with Angelica in that "me duele la garganta" is more specific, thus "more correct".
But the fact that speakers (particularly in Spain) will use "me duele el cuello" as a natural way to say "I have a sore throat" is a fact. Metonimic if you wish (el todo por la parte) but nonetheless used in day to day speech.
If I say "tengo un dolor de garganta de narices" means I have quite a sore throat, although not a problem with my nostrils...
If I say, "me duele el esternocleidomastoideo" then it is specifically the muscle, and never would be understood as the "throat".
"Me duelen las anginas" "Me duele la laringe" "Me duelen la amígdalas" would be expressions I've heard too, but the last 2 may be a bit highbrow... or rather pedantic...
Like "tener un trastorno catarral del tracto respiratorio" would be going overboard...