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.