The curious case of the missing Coat of Arms...
This is quite independent of the fact that Nikolai's Birthday is not a flag day in Denmark. Queen Margrethe II has decided that only those family members in line of succession and who support her in her role as head of state are on the official flag days' list. These are her two sons, Crown Prince Frederik and Prince Joachim, as well as their wives, Crown Princess Mary and Princess Marie (NIkolai's step-mother), and the Queen’s sister Princess Benedikte. Birthdays of children of the Crown Prince of Denmark, may become official flag days when the Prince or Princess has turned 18, and if the monarch so wishes.
Grandchildren of the British monarch are assigned a Coat of Arms to celebrate their coming of age at 18. The next should be Lady Louise Mountbatten-Windsor in November of next year (when she will also decide officially whether to be known by her legal title which is HRH Princess Louise of Wessex or remain Lady Louise). Nikolai is already 21.
In 2018 Nikolai started working as a fashion model and last year he was admitted to Copenhagen Business School to study for a Bachelorate in Business Administration and Service Management. Nikolai is not expected to carry out official royal duties, although he is the recipient of a number of commemorative medals and has attended state banquets. He has not yet been appointed a Knight of the Order of the Elephant.
The same year, the Chapter of the Royal Orders of Chivalry in Copenhagen confirmed that no decision had been taken about the Coat of Arms of the Prince. It may be the case that Nikolai simply does not require a Coat of Arms and has little opportunity in his modern life to make use of one. His Arms have not been registered or he has simply asked not to be assigned any. The same goes for a Monogram which the Scandinavian royal families make great use of and which, in Denmark, are designed by The Queen herself being an a very accomplished artist.