The UK has a monarch, but the 'goverment' is elected. This is different to an absolute monarchy, where the king or queen makes the laws.
royal monarchy
A country run by a royal family would be described as a monarchy.
The old British model of government, where the king and queen had political power
The British Royal Family.
Royal government is a Monarchy or where the head or titular head is a King Queen or Emperor.
This is the description of a monarchy.
Monarchy
A monarch is the sole leader of a monarchial government and that type of government is also called a monarchy
The United Kingdom is a monarchy. Until the revolution, France was a monarchy as was (and some still are) most of the countries of Europe. Any civilization with discrete boundaries and a royal family was/is a monarchy...example Ancient Egypt. There has to be a single king or queen.
Governments never use Royal families to govern. A Constitutional Monarchy allows the Monarch to advise and to warn the elected government.
This is the description of a monarchy.
An example of metonymy is when someone says "the crown" to refer to the monarchy or royal authority. In this case, "crown" is being used to represent the power and institution of the monarchy.