That sounds weird. It's pretty easy to get those if you want to marry, but her being your girlfriend shouldn't help outside of the formal letter of invitation.
One doesn't need to be married in order to be covered by the EU rights in question. Straight from the source:
"The following family members of an EU/EEA citizen or Swiss citizen, who is exercising his/her right to free movement in Denmark, or of a Danish citizen who is exercising or who has exercised his/her right to free movement in another EU/EEA country or Switzerland, has the right to have a visa application processed in accordance with EU regulations:
"The following family members of an EU/EEA citizen or Swiss citizen, who is exercising his/her right to free movement in Denmark, or of a Danish citizen who is exercising or who has exercised his/her right to free movement in another EU/EEA country or Switzerland, has the right to have a visa application processed in accordance with EU regulations:
* spouse
* cohabiting partner aged 18 or over <----
* ..."
See https://www.nyidanmark.dk/en-us/coming_to_dk/visa/visas-unde....