The major similarity between monarchy and democracy is that they are both forms of government. Government is dedicated to establishing hierarchy of authority, rule of law, social order and security. With this comes government taxation, government military needs, government assistance for the poor and disabled (e.g., workhouses, pensioners homes, Welfare and food relief).
Monarchies are limited, also called constitutional, or absolute. In limited, or constitutional, monarchies there is a division between the governing body and the ruling body so that an independent body, usually a parliament, establishes laws, while the ruling body, the monarch, remains influential in affairs of state and public welfare. With the monarchy limited by a constitution, as Great Britain's monarchy is, the role of the monarch is defined constitutionally.
In absolute monarchies, except for monarchical advisers, there is no body other than the monarch his- or herself who makes laws and sets the governmental taxes. One thing that confuses the roles of a monarchy is that lesser heads of state, such as dukes and earls, originally had overlapping powers, such as to raise an army and to levy taxes on the people living on their land (once called peasants), but this overlap was on a lesser scale and only mimicked the absolute authority and power an absolute monarch has. Modern-day absolute monarchies exist in Saudi Arabia and in United Arab Emirates.
Monarchs are not elected nor are they appointed. Monarchs gain their power, during a peaceful succession, by hereditary right. Generally, the crown of a monarch is passed to the eldest son upon the monarch's death or abdication. If there is no son in line of succession, the crown may go to the nearest relative, sometimes to a girl, as in the case of England's Queen Elizabeth I, but often to a male relative.
During a militant succession, in which the right to rule is contested or challenged, as was usually the case in the ancient Scottish monarchy, the crown of the monarchy is passed to whoever is most powerful in the battle that ensues over the right to rule. It is during these militant successions that new dynasties or forms of government are introduced, as when Cromwell was a signature to the beheading of Charles I (January 1649) and the monarchy was deposed altogether to be replaced with short-lived republican rule in the Commonwealth of England.
Democracies are governed by elected heads of state, usually a President or a Prime Minister. The right to make laws, levy taxes, raise a military and engage in war resides in the legislative bodies, a parliament or a congress, that have also been elected by the populace. The president or prime minister of a government works in tandem with the legislative bodies, while the court system tests and validates or challenges laws and actions that are called into question. The paramount difference between an absolute monarchy and a democratic (and/or republican) government is that there are safeguards in place in a democracy so that any action taken by a head of state or a legislative body can be challenged in the highest court and, possibly, rescinded. The flaw comes in that courts cannot ultimately be challenged; when the highest court gets it wrong, it usually stays wrong.
Power is transferred in democracies in peaceful elections that may usher in new leadership and, possibly, new ideology, as when a Communist wins an election instead of a republican in a South American democracy. It is true that the process of peaceful democratic election can be thwarted if certain groups join together to overpower or corrupt the process. This may happen in the case of a military coup or in the case of election tampering and/or intimidation. Some historians hold to the idea of election tampering in Florida during the 2000 presidential election pitting George W. Bush against Al Gore.
Unless we are talking about a constitutional monarchy like that of Britain, there is very little similarity between a democracy and a monarchy. Britain long had a monarchy in which there was a great deal of democracy. That, however, was relatively unusual.
A true monarchy has little in common with a democracy. In a monarchy, the monarch is not responsible to anyone. Monarchs are not elected. They do not have to have elected parliaments that make laws and that convey the will of the people to the government. A monarchy is really an autocratic form of government while a democracy is based on the will of the people.
Of course, there are many kinds of monarchies. The British monarchy, especially after the Glorious Revolution, did have a Parliament that was elected and that brought a great deal of democracy into the system. But this is not true of all monarchies.
A monarchy is typically a form of government that is very different from a democracy.