The major similarity between monarchy and dictatorship is that they are both ruled by one person who is not answerable to anyone else. Both of these are fundamentally undemocratic forms of government.
In a monarchy, the person who rules the country rules because they come from the right family. Monarchies generally pass the rule of the country from parent to child within a family. That is why, for example, Queen Elizabeth I of England came to power. She became queen because her father, Henry VIII of England, had been king. In a dictatorship, it is harder to know how power will be passed down. It can be passed from father to son, as in North Korea today. It can also be passed down in other ways, as it was from Josef Stalin to Nikita Khrushchev in the Soviet Union. The similarity, though, is that these rulers are not chosen through any sort of democratic process.
Another important similarity is that whoever has the power in both these systems is truly in charge and is not answerable to anyone else. The President of the United States has to answer to many people. He (or someday she) cannot do much unless Congress agrees. This is not the case in a true monarchy or a dictatorship. The rulers in both of these systems are able to simply declare what the law will be. No one can overrule them and they do not need anyone’s consent in order to pass laws.
Thus, while these systems have their differences, they are similar in that their rulers are not elected democratically and do not have to answer to anyone or get anyone's permission to make laws.