Strange, if he needed to sign again (which in itself doesn't make a lot of sense, unless he has to promise something different than before), why didn't he have to do so before he first acted as Regent? Almost a year and a half later seems rather late and random.
The short answer is that Christian would have needed to repeat his promise to observe the Constitution anyway, so he may as well do it now.
Two different laws are involved: the Constitution and the regency law.
The Constitution of March 27, 1953, Article 8, requires the King to promise to observe the Constitution as a precondition for conducting the government. Note that it says the King is permitted to make this declaration ahead of time, while he is the Heir to the Throne (first in line).
“§ 8
The King shall, prior to his accession to the throne, make a solemn declaration in writing before the Council of State that he will faithfully observe the Constitutional Act. Two identical originals of the declaration shall be published, one of which shall be delivered to the Folketing to be kept in its archives, while the other shall be filed in the Public Record Office. When, because of absence or for other reasons, the King is unable to sign the aforesaid declaration immediately on his accession to the throne, government shall, unless otherwise provided by statute, be conducted by the Council of State until such a declaration has been signed. When the King has already, as Heir to the Throne, signed the aforesaid declaration, he shall accede to the throne as soon as it becomes vacant.”
The Law on the Conduct of the Government in the Event of the King’s Minority, Illness or Absence of February 11, 1871 requires any appointed regent (i.e., any substitute except the Heir to the Throne, who steps in automatically and does not need to be appointed) to make the same promise as the King to observe the Constitution as a precondition for conducting the government:
“§ 6
No one shall take up the conduct of the Government as Regent until he has made a sworn assurance such as is stipulated in Article 7 of the Constitution.”
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Note: As the regency law dates to 1871, it refers to “Article 7” of the old Constitution of 1849, which was fundamentally the same as Article 8 of the current Constitution of 1953, quoted above.
So it is the same declaration. However, the “problem” for Crown Prince Christian is that the Constitution states “When the King has already,
as Heir to the Throne, signed the aforesaid declaration, he shall accede to the throne as soon as it becomes vacant”. Thus, the declaration that Christian signed when he was second in line to the throne,
before he became the (immediate) Heir to the Throne, would not count once he became King, and he would need to repeat the same declaration at that time. So one way or the other, he would need to declare his loyalty to the Constitution again.