Under international law the Vatican City is a state. It is distinct from the Holy See, the episcopal jurisdiction of the Catholic Church in Rome. The Bishop of Rome enjoys a primacy over other Catholic bishops and is Pope.
Unlike the Vatican City, the Holy See is not a state but has long enjoyed international legal personality- longer than almost all modern states and longer than all other organisations enjoying international legal personality. Ambassadors including Australia's are accredited to the Holy See. The Holy See is a state party to a number of international conventions or treaties, and enjoys permanent observer status at the United Nations.
The Vatican City is an elective monarchy, in which the head of the Catholic Church, the Pope or Supreme Pontiff, is the Sovereign. As such His Holiness enjoys the fullness (or plenitude) of legislative, executive and judicial powers (Article 1: Fundamental Law) .
The Vatican City is thus an elective and absolute monarchy, one which is obviously not hereditary.
…republic & head of state…
This example of monarchy demonstrates the fallacy in a constitutional debate of attempting to limit the meaning of words so as to fit a political agenda. We see this in the attempts of those wishing to remove the Australian Crown of imposing their agenda driven restrictive meaning of the word “republic”.
To describe the Commonwealth Australia as a republic, albeit a crowned republic, is not erroneous. The term “head of state” is somewhat different as it already has a technical meaning in customary international law.
Those who want significant constitutional change – even if it is negative rather than positive – will not persuade the Australian people by what is no more than a Humpty Dumpty attempt to limit the meaning of words to fit their agenda. Unwilling to reveal their plans for the constitution and the Flag, like Humpty Dumpty, they wish to be the masters.
'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it means just what I choose it to mean — neither more nor less.
''The question is,' said Alice, 'whether you can make words mean so many different things.
''The question is,' said Humpty Dumpty, 'which is to be master — that's all.'
(Through the Looking Glass, by Lewis Carroll)
In Australia, the people are the masters, as we saw in 1999.