When it comes to the "monarch - subjects relationships", there has historically been a lot of variety, depending on the part of the world. Asian rulers are often deified (for instance, Japanese Emperors are believed to be direct descendants of the Supreme Goddess Amaterasu, while criticism against the king in Thailand is considered sacrilege and punished by law). Eastern European monarchs were sometimes seen as God-given parental figures whose right and duty was to take care of and, mostly, punish their usually unruly and sinning "children"(Russian kings/queens would be traditionally addressed as, what can be roughly translated as, "Little Father"/"Little Mother"). Western European royals are rather perceived as legal entities that must fulfill their contractual obligations to keep enjoying a considerable number of rights and perks (this interpretation of monarchy might have been introduced as early as 1215 when King John of England was essentially forced to sign a Charter of Liberties known as Magna Carta).
Danish warm and friendly attention to and curiosity about their monarchs has made sure that none of the above would apply! The best way to describe a unique relationship between Danish kings and their subjects might be to compare the former to...their namesakes on the chess board! In chess the king is the most important but also the weakest (at least, for most part of the game) piece of the whole set. In Denmark the monarch is the mirror of the country's international image and the primary symbol of its unity - and yet, his power is symbolic, as well, because key decisions affecting the country's daily life are made by the Parliament. Besides, as most public figures, the royals have very little privacy. It's therefore not surprising that they are sympathized with, and sometimes even spoiled. For instance, several lovely rotunda pavilions were built for the royal family right on the waterfront (and not far from the Mermaid) - so that Queen Margrethe II and her kin could rest comfortably after yachting for a few hours. Even the police would occasionally turn a blind eye to royal misdemeanors - just like it did when Crown Prince Frederick decided to swish through Copenhagen at 300 km/hour. Usually, the rules are observed much more strictly: when the visiting Swedish king followed suit, he was fined an equivalent of 750 $ - even though his speed merely reached 200 km/hour...