If anything it is the other way round. The key political difference between Sunni and Shia Islam is the status of the family of the prophet.
In Shia Islam, God chose Ali, who was Muhammad's cousin and son-in-law, and Muhammad's closest blood relative as the leader of Muslims after Muhammad. The leaders of Shia Islam claim a direct bloodline to Ali.
In Sunni Islam, following the guidance of Muhammad's actions in life, his father-in-law Abu Bakr was elected Caliph.
The Sunni branch of Islam stipulates that, as a head of state, a caliph should be chosen by Muslims. Followers of Shia Islam, however, believe a caliph should be an Imam chosen by God from Muhammad's direct descendants.
However this applies to the Caliph, the leader of Islam. There is no widely recognised Caliph in Islam, on either the Shia or Sunni branches. The various Kings and Emirs of Islamic countries do not claim the Caliphate. The Leader of Iran doesn't either. The leader of Daesh claimed the caliphate, but this claim was not widely recognised.
As such there are many Kings and Emirs of Islamic countries, while other countries have a theocratic leader, and other are republics. There are a number of notable Sunni Monarchies, but other Sunni states (Egypt, Pakistan for example) are Republics.