It's more fun to say. Quite seriously.
It also sees use in popular culture like The Crown, which must in itself have gotten the idea from somewhere.
Glücksburg is a moated castle in today's Germany, near Flensburg. It was build in 1580-87 by the third son of king Christian III. of Denmark, Johann the Younger (in Danish: Hans den Yngre) whose brother Frederick II. gave him lands and formed the duchy of Schleswig-Holstein-Sonderburg for him.
The Royal family at that time were the former reigning Counts of Oldenburg and dukes of Schlewsig-Holstein, who had inherited Denmark's throne after the former (male-line) Royal family had died out and they were the next via the female line in descent from king Valdemar the Great). First king of that line was Christian I. The family held the lands of Schleswig and Holstein under Danish rule till the Prussian won the German-Danish war in 1864 and Schleswig-Holstein became a German Federal State in 1946, but the titles of the family exist till today (as names only in Germany!)
When Hans den Yngre died in Glücksburg Castle he left the castle with part of his lands to his fifth son Philip, who became the first Duke of Schleswig-Holstein-Sonderburg-Glücksburg. When this line died out, the title fell back to the Danish king.
In 1825, a prince from the Danish Royal line of Schleswig-Holstein-Sonderburg-Beck (another of the small duchies Hans den Yngre had parted his lands into, left to another of his sons), Frederik William, married the sister of king Frederick VI.'s queen. To celebrate that, his brother-in-law created the prince as Duke of Schleswig-Holstein-Sonderburg-Glücksburg. That's the line the Royal Houses of Denmark, Norway, Greece and now the Uk all descend from.
Because due to the close relation and contact, when again the Royal male line was about to die out, the Danish parliament of that time elected the Duke Frederick William's son Christian as heir presumptive in 1853, he became king as Christian IX. (He is the king who was thought of as the grandfather of Europe's monarchies).
So even though the Royal families all prefer the shortened name of "Glücksborg" for their dynasties, it is in fact a minor line of the Danish Royal House who reigned the SHSG-duchy, which became the male-line when the elder lines died out. And the long name is because in German nobility, minor lines take the territorital destination of the main line they are descended from and add the name of the little piece of land they inherited as part of the name to distinguish them from other lines. And as older and more major lines died out, even a prince of SHSG had a chance to be voted in as the new king as he was related to the late kings of the country in the paternal line.
Today, though, with no more death through diseases of the children and no more deaths of Royals in military service, the chances that Prince Joachim's children are needed for the throne are completely different from these times.
Hope this helps!