Not to nit-pick, but you probably meant "a Dane" :-) "A Danish" is usually a form of pastry you get at the bakers in English speaking countries. (There was no audio in this form of the question so I can't really address your original question, but if I ever come across it in a form with audio I'll be sure to add my opinion. In general many of the audio clips are poor so don't despair!)
You're totally right. I actually knew this, but in my first language (French), the pastry and the person look much closer (un danois (person) - une danoise (pastry)), so I tend to forget to use the word Dane when I don't think about it... what do you mean there is no audio in this form of the question? The text is given and you can listen to it. The question is not about the audio but still...
These "forums" apply to any place the sentence is used. In my case I had two words presented and then a drop-down to select "officielt" among 3 or 4 choices. There was no audio in that case. You must have had some other version of the exercise with the sentence "Ægteskabet er officielt" that had an audible part (as you mention the fast voice)?