Islam (Arabic : اسلام) is a religion started in Saudi Arabia, and who invited people to Islam, was Respected Prophet of Islam, Muhammad, son of Abdullah, known as Prophet Muhammad (SAW). Islam has two large branches, Shi'a (Can be found in Iran, Iraq, Syria, Azerbaijan, Libya and Tunisia) and Sunni (almost rest of Ummah). Islam has a good connection with monarchy, as other religions do.
The perfect state and utopia according to religions (even very liberal ones, like Buddhism and Baha'i faith) is monarchy. Monarch may be called Pope or something else. In this article, you know Islamic models of monarchy more.
If you want to know writer's opinion about this, I must say no, but why? Mahdi (means "The directed one" in Arabic), is the Messianic person of Islam. In Twelver Shi'a belief, he's 12th Imam, in Sevener (Ismaili) belief, He's Jesus and Iranian Sunni people believe that a person will be born, and spirit of God (Ruhollah) will be with him, Ruhollah means "Son of God" (Not biological), too. This form of monarchy was popular in Sudan, and no other official state in this form was available.
Caliph (Arabic : خلیفه) means Representative of God , and a nation ruled by Caliph called Caliphate (Arabic : خلافة). Caliphates are in great empire sizes. Notable caliphates are Rashidun (Abubakr, Omar, Uthman, Imam Ali (A.S)), Abbassid Caliphate in Sunni world and Fatimid Caliphate in Shi'a world.
Imam means leader in Arabic. So, it's a nice Islamic leadership. But, Imamate can be a Sovereign order with moral power, like Aga Khani Ismaili Shi'a people . But there's differences in definition of Imam. Sunni people claim people who have book as Imam, Twelver Shi'a claims Dhuriah (Direct generation of prophet Muhammad) as Imams, and Ismaili people claim people who have a connection to Muhammad's generation as Imams (for example, my ancestry from mother is Prophet Muhammad). Imamate can be as large as Persian empire, or as little as current Malta!
Sultanate is only Kingdom/Empire with Islamic and Arabic culture. Also, some Turkic cultures accept Sultanate. Current Sultanates are Oman (Arab), Brunei Darussalam (Asian). Former Sultanates were in Turkey, Arabia, etc.
Emirate is an Islamic Principality. A little nation, or part of other nation with a prince called Emir. Current Emirates of the world are UAE, Qatar and Kuwait.
A normal person who has no roots in Muhammad's generation, but knows enough about Islam and Islamic concepts of government called Sheikh. So, his nation called Sheikhdom. Lots of Arabic cities and sub-national entities are Sheikhdoms.
Dervish is a Shi'a term, and it's used specially in Twelver Shi'a culture. Dervish means "Devotee of Imam Ali (A.S)" and a Persian word. The only Dervishate were Safavid dynasty, a Persian and Turkic government ruled Persia. Dervishate is like Imamate, can be as great as an Empire or as little as Vatican city!
These are Islamic monarchist systems. Also, there's another choice called Islamic Republic and common in Asia and Africa. Iran, Afghanistan, Pakistan and Mauritania are current Islamic Republics of the world. Islamic republic can have concepts of one of systems above, but it's still more democratic than them.
I will write about other common systems in Asia/Middle East/Africa as soon as possible.