https://www.kongehuset.dk/en/news/hm-the-king-appointed-as-admiral-and-general
https://www.kongehuset.dk/nyheder/h-m-kongen-udnaevnt-til-admiral-og-general
On the occasion of his accession to the throne, HM The King has been appointed to the highest military rank in the Army, the Navy and the Air Force respectively. As admiral and general, His Majesty is now of the same rank as the Chief of Defence.
[...]
Without knowing much about the Danish military, I am assuming from the excerpt above that Denmark is similar to Australia and Canada, where the Chief of the Defense Staff (in Denmark, the "Chief of Defence" using British spelling) is the
only active military officer who holds what in the United States is called a "four-star rank", while the commanders for example of the Army, the Navy, and the Air Force, who serve under the CDS, hold the equivalent to a three-star (maybe two-star?) rank only. Maybe Muhler can confirm that later.
Note that this is not the case in the UK, where both the CDS, the Vice CDS and the 3 service chiefs (plus one or two commanders like the Commander UK Strategic Command and the Deputy Supreme Allied Commander Europe) now hold (the equivalent to) four-star ranks while the King actually holds higher "
five-star" honorary ranks (in the UK, Field Marshal in the Army, Admiral of the Fleet, and Marshal of the Royal Air Force). The British CDS also used to hold a five-star rank, but was downgraded to four-stars in the late 1990s, I think.
Note 1: in connection with the UK, I am using the terminology "xx-star" just to provide an international comparison based on US ranks, but, strictly speaking, British general officers do not wear "star" insignia.
Note 2: In the United States, of course, there is an insanely high number of four-star general officers by international standards (I think, currently 44), but the US military is huge with multiple combatant commands and headquarters.
EDIT: I just checked on the Wikipedia and it looks like 3-star ranks like Lieutenant General or Vice Admiral, which, in Australia or Canada, are held both by the Vice CDS/CDF and the service chiefs, are reserved in Denmark to the Vice Chief of Defence only, while the service chiefs hold only two-star ranks (Major General or Rear Admiral).