Mostly, the wives of Danish royal princes have been styled "Princess (own name)", but around the early 20th century, there was a period of time when the form "Princess (husband’s name)" was also used.
In addition, King Christian X (reigned 1912-1947) did not permit wives who were not of royal birth to use “Princess (own name)”, but granted them the title of “Princess (husband’s name)”. (He also demoted their husbands from "Prince to Denmark" to plain "Prince" for marrying them.)
Below are some historical instances of Princesses consort being referred to with their own first name.
From the late 18th century:
The following is taken from a list of historical documents held by the royal library of Copenhagen concerning Sophie Frederikke (1758-1794), the wife of King Christian VII’s half-brother Hereditary Prince Frederik. All of the document titles refer to her by her own name.
The list is long, so I only quote the first few entries, but the full list may be found on page 459 of Volume 3 of the catalog, which can be downloaded
at this link.
Treschow, Herm., Nogle faa Traek af Hds. Kgl. Høih. Arve-Prindsesse Sophie Friderikke efter Høistsammes Død den 29. Nov. 1794. Kbh. u. A. P.
Rothenburg, Mort. Trane, Ved Hds. Kgl. Hoih. Princesse Sophie Friderieces Geburtsdag den 24. Aug. 1774. Kbh. 4. 2 Bl. [Digt.]
Bano, Chr. Frid., Til Hds. Kgl. Høih. Prindsesse Sophia Friderica paa Høistsammes Fødsels-Dag den 24. Aug. 1783. U. St. Fol. 1 Bl. [Digt.]
Bero, Sophia, Underdanigst Lykenskning til Prindsesse Sophia Friderika, paa Høistsammes Fødselsfest den 24. Aug. 1784. Kbh. Fol. 2 Bl. [Digt.]
From the late 19th century:
The
Royal Court and State Calendar of 1880 refers to princesses consort with their own names:
Protectrice.
Hds. Kongelige Hoihed Arveprindsesse Caroline. [Page 663.]
Kronprindsesse Louises practiske Tjenestepigeskole.
Hendes Kongl. Høihed Kronprindsessen er Skolens Protectrice og har naadigft tilladt, at den bœrer hendes Navn. [Page 759.]
From the mid-20th century:
The official announcement of Princess Margrethe’s birth in 1940,
quoted on the Royal House website, refers to her mother by her own name:
Den 16. april 1940 ændrede dog karakter, da en lille pige på 3.300 gram kom til verden klokken 10.10 i Frederik VIII’s Palæ på Amalienborg. Efter kun 23 minutter blev følgende telegram udsendt: ”Hendes Kongelige Højhed Kronprinsesse Ingrid nedkom Tirsdag Kl. 10.10 med en velskabt Datter. Moder og Barn befinder sig vel.”