I agree, 'His girlfriend is Dutch' also sounds more natural to me. I think the reason for the translation is that the English adjective 'Dutch' can be used for both genders, while there is a distinction in Dutch. The Dutch sentence thus contains information that would be lost by using 'Dutch' rather than 'Dutchwoman', but the 'woman' part in this sentence is already included in 'girlfriend'.
But this translation, "His girlfriend is Dutch" is grammatically different than the original sentence, because "Dutch" is an adjective here, whereas "een Nederlandse"="a Dutchwonan" is a noun. I have the impression that using the noun form for a person in not polite in English anymore, but I am not a native speaker. It definitely seems common in Dutch.
It can be used for both meanings, but zijn vriendin usually will be girlfriend. Because of this ambiguity quite a lot of people (like myself) prefer to use een/deze/die vriendin van hem when friend is meant.
The fact that gay relationships are getting more normal and acceptable in everyday use (yes even in the Netherlands we're not there yet), means that Zijn vriend is een Nederlander. or Haar vriendin is een Nederlandse are getting the same ambiguous meaning, these used to simply mean friend.