There are more differences than similarities. A monarchy relies on the rule of one person, while a democracy rules through a consensus of the people. Even with this difference, there are some commonalities. Many monarchies have claimed to have ruled in the name of God, while many democracies have claimed to rule in the name of the governed. In other words, the leaders of both forms 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.