The commonwealth is also a collection of countries ruled by the crown; but it's not a country whereas the UK is a country.
reply1) Most Commonwealth countries are republics, not monarchies.
reply2) The monarchies in the Commonwealth countries are ruled by different crowns; but, for historical reasons, most of those crowns are worn by the same physical head. (But not all of them. The King of Malaysia is a different natural person from the King of the UK, for example.)