Actually, I think that you'll find that though she lived in Copenhagen she was actually born in Tveje Merløse parish in Holbæk Amt. I found a 'Politietsregister' for 'Caroline Anna Cathrine Mortensen born Aug 15 1875. (It shows addresses where she lived in Copenhagen from 1891 through 1896, and gives her birthplace as Holbæk.)
I looked in the parish register for Tveje Merløse and found her birth/baptism record there. Her parents were Anders Mortensen and Ane Kirstine Jensen (who was 29 y.o. at the time of Caroline's birth). The 1880 census for Tveje Merløse shows she had a sister named Birthine Sophie Mortensen (age 6 in 1880) and a younger brother named Anders Christian Mortensen (age 3 in 1880).
Hope this helps,