It is not a political system. Like any other church, even down to the Unitarians, the Catholic Church has its own internal laws and its moral laws.
The Catholic Church has never been the same as the civil ruler, and in fact, the reason that many Catholics died was because of the conflicts between the civil ruler and Church authorities.
Henry VIII tried to establish himself as both the civil and religious ruler, but that didn’t work because Christianity can’t work that way, so all he really did was make Anglicanism the state church with the king or queen of England as its head.
The Catholic Church has NO civil power. The only place it has any power is the Vatican, which is a tiny fictional state composed mainly of a handful of clergy and religious that was set up to defend Catholic sites - such as St Peter’s Basilica - from secular powers that were trying to absorb or even destroy them.
It certainly sounds both political and religious to me.