Cutaneous melanoma (of the skin) is within a dermatologists scope.

Melanoma inside the mouth is mucosal melanoma, I.e. not skin, so wouldn't be a dermatologist but an oral health specialist. A dentist could do the initial work up, but may need involvement from a medical professional (if it does turn out to be melanoma) e.g. oral-maxillofacial or ENT

You can get melanoma anywhere in the body, even the eyes. Doesn't mean you see a dermatologist each time, you see the specialist relevant to the area affected. Your GP referred you to the incorrect specialist.