The grammar term for something like this is a "Demonym" or "a word that denotes the members of a people or the inhabitants of a place."
For Denmark, it is irregular.
Demonyms take two forms 1: Adjective 2: Noun
Some demonyms are the same for both. For people from Denmark, the following is true:
Adjective: danish. Describes a noun. "The Danish government"
Noun: a person in this case. "Per is a Dane from Copenhagen."
Think of it in this regard:
Denmark/Danish/Dane is like Spain/Spanish/Spaniard
So in that case, a 'person that is from Denmark' is a Dane.