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.