of government seek some sort of legitimacy outside of themselves.

Most monarchies are hereditary. While democracies allow people from all walks of life to be leaders, there are tendencies to favor leaders with strong family ties to governing. Both democracy and monarchy were praised and demonized in Plato's Republic. According to Plato, a monarchy could be good if it had a good monarch, but bad if it had a tyrant. A democracy could be good if it responded to the will of the people, but it could turn into anarchy if everyone acted selfishly.

Though democracy and monarchy have more differences than similarities, there are some ways in which they overlap. Both democracies and monarchies govern through a consolidation of power. In a democracy, this power base is democratically elected by the masses; in a monarchy, the power base may be appointed through any number of means, but typically through primogeniture. In a democracy, the majority rules, while in a monarchy, only one person rules. However, even absolute monarchies throughout history often consulted a trusted group of advisors in order to gauge the popularity of proposed or enacted policies, help him or her understand the political ramifications of their actions, and let them speak for the common people. It's important to remember that, while monarchs can be tyrants, monarchies and dictatorships are not the same thing. The majority of present-day monarchs either have no state powers, or share power with democratically elected governments. Fewer than a dozen monarchies remain in the world; most are in the Middle East. Vatican City is also considered an absolute monarchy, with the Pope as the head.

The similarities between monarchy and democracy depend upon the monarchy and the democracy in question.

If you want to talk about the similarities between America’s current democracy and monarchies in general, you could probably discover the two have quite a bit in common.

One main similarity involves the power to declare war or involve their country in a war. Many monarchs had the power to wage war. Prussia’s king, Frederick II, led Prussia into more than one war against Austria. Likewise, Louis XVI, France’s king, had the power to provide support for American colonists during their war of independence against British.

Presidents, too, seem to now have the power to declare war in America’s democracy. When America’s democracy was first designed, Congress was supposed to have the power to declare war. However, after the attacks on September, 11, 2001, you could argue presidents now have the de facto ability to wage war.

Barack Obama involved America in a war with Libya without Congress’s approval. His successor, Donald Trump, appears to have also espoused the right to wage war without Congress’s approval. When four US troops were killed in Niger in 2017, many senators admitted that they didn’t know America was fighting in that West African nation.

Another similarity between America’s democracy and general monarchies is the importance of family ties. With monarchs, the position of ruler typically stays in the family. When Louis XV died, his son, the aforementioned Louis XVI, became king. Likewise, in England, when Mary died, her sister, Elizabeth, became queen.

In democracy, family ties seem to have a central importance. Before George W. Bush was president, his father, George H. W. Bush, was president. George W. Bush’s brother, Jeb Bush, also tried to become president.

Then there’s Hillary Clinton. Her marriage to Bill Clinton appears to have led her to try to become president. There's also been periodic encouragement for Michelle Obama, Barack Obama's wife, to consider a presidential run.

In The Napoleon of Notting Hill, G. K. Chesterton argues that the most absolute of monarchies has a democratic aspect. The monarch is a representative of the common man in the sense that he is a random sample of humanity. A president or prime minister must be ambitious, perhaps clever, perhaps unscrupulous to attain that position. A king becomes a king simply because his father was a king, and may have any combination of qualities. He is far more likely to be an ordinary person in terms of intellect and character, and therefore to understand what ordinary people want, than the type of person who is prepared to spend decades building a political career.

Chesterton uses "democracy" as an attitude of mind, almost a spiritual state, rather than a technical description of a political system. Absolute monarchies are technically the opposite of democracies. However, the experience of Europe over the last few centuries has been that absolute monarchies tend to transform, with varying degrees of violence, into constitutional monarchies as societies develop. The modern constitutional monarchies of Britain, Spain, the Netherlands, Sweden and other European nations are essentially identical to democracies. The monarch performs some (mainly ceremonial) duties, which in other democracies are the responsibility of a president, but there are no substantive differences from other democratic countries.

A true monarchy and a democracy are more notable for their differences than for their similarities. Democracy means that citizens make decisions for their nation or elect politicians to make these decisions on their behalf (as in representative democracies). These government officials and politicians serve limited terms of service before stepping down or being reelected. Monarchies are usually ruled by a dynasty of rulers who come to power through the right of heredity. Monarchs serve for life. In an absolute monarchy, the people have little, if any, say.

There are, however, constitutional monarchies in which the monarch has limited power or merely serves the function of a figurehead. In this case, the country functions as both a democracy and a monarchy. In this type of monarchy, people elect representatives to a legislature in much the same way that people do in a republic. Both systems have courts to settle legal matters. In a representative democracy, judges are elected by the people or appointed by elected officials. This is true in most constitutional monarchies as well. In more traditional or absolute monarchies, judges are appointed directly by the monarch or the monarch serves this role themselves.

As you can see, there are some similarities between a constitutional monarchy and a representative democracy. An absolute monarchy is another matter altogether, as there are few, if any, similarities to democracies.

I believe your question is focusing on a constitutiinal monarchy as in a true monarchy there aren't many similarities.  There are similarities between a constitutional monarchy and democracy.  In both systems, the people elect their leaders.  While in a constitutional monarchy the King or Queen isn't elected, the lawmakers and the head leader such as a Prime Minister are elected.  In a democracy, all leaders are elected.  This includes the lawmakers and the chief executive.  In both systems there is a system of justice.  Laws and courts are created to ensure justice.  In both systems, people are able to influence the legislative and political process.  While the will of the people can be more clearly expressed in a democracy because there is no unelected King or Queen, the will of the people can be expressed and acted upon in both systems.  There are several similarities between a democracy and a constitutional monarchy.

The main similarity between monarchy and dictatorship is that they are both ruled by one person who does not have to do what anyone else says.  This means that both of these forms of government are very undemocratic.  (Please note that I am talking about real monarchies, not constitutional monarchies where the monarch is only a figurehead, like in the United Kingdom today.)

In a monarchy, the monarch gets to rule simply because of who their parents were.  In a monarchy, the right to rule the country generally passes from a ruler to their child when the ruler dies.  As an example of this, Henry VIII became King of England when his father, Henry VII, died.  In a dictatorship, power sometimes is passed from parent to child.  One example of this is North Korea, where power passed through three generations of the Kim family.  Not all dictatorships hand down power in this way, but it can happen.  What is always similar between the two systems is that leaders are not chosen democratically in either one.

The other main similarity between these two systems is that whoever is in power has essentially complete control over the country.  This is very different from our own system.  In our system, the President of the United States cannot do very much without the consent of Congress.  The president is answerable to other people.  In a dictatorship or a monarchy, leaders do not answer to other people.  Leaders in these systems are able to simply declare what the law will be and do not have to get anyone else’s consent.

The main similarities, then, are that both of these systems have rulers who are not elected and who enjoy essentially absolute power.