First, to correct astrochuck, Joseph Smith (founder of Mormonism) not only allowed blacks into full membership, but also extended priesthood offices to them. Elijah Abel, a black Mormon in the first years of the church, eventually became a member of the Quorum of Seventy, a presiding body of elders. The policy that banned certain groups of blacks (specifically pure-blooded African descent, not aboriginal blacks of other regions) surfaced in pioneer Utah. Astrochuck is correct in stating that priesthood ordinations were officially extended to black males in 1978. To double check my facts here, reputable historical/sociological works like Richard Bushman’s “Joseph Smith: Rough Stone Rolling” and Armand Mauss’s “All Abraham’s Children” will provide ample primary source material.
Now, to address the question at hand (what does Mormonism have in common with Islam), first, both faiths recognize a kind of prophetic authority. Muhammad claimed to have been visited and taught by the angel Gabriel; Joseph Smith claimed several visions, including God the Father, Jesus Christ, angel Moroni (a Book of Mormon character), Elijah, John the Baptist, apostles Peter, James, and John, and many more.
Antebellum Americans picked up on this and other similarities, and went so far as to discredit Joseph Smith’s claims with a guilt-by-association attack with Muhammad (see J. Spencer Fluhman, ”‘An American Mahomet’: Joseph Smith, Muhammad, and the Problem of Prophets in Antebellum America,” Journal of Mormon History, Summer 2008.
Islam holds to rituals of sacrifice and pilgrimage that resemble some concepts in Mormon theology, but it’s safe to say that the two are very much distinct and nuanced more than compellingly similar.
Mormon concepts of polygamy actually have roots in a unique doctrine of Celestial marriage that came onto the scene with Joseph Smith, though Smith died too soon to announce it and teach it to the whole church body. Brigham Young was the main figure in leading the pioneer church into polygamous practices. The 4th president ended polygamy in the 1890s after the United States government passed the Edmunds-Tucker Act, a law that disenfranchised the church and its property (temples, among other things, immediately became property of the state and were repossessed by the federal government). Many Mormons at that time disagreed with the decision to abandon polygamy, and many (most) fundamentalist polygamy groups that have come out of Mormon can trace their beginnings to those that fractured off from the main church body at that time.
Islam on the other hand does have doctrines relating to polygamy, though of a completely different nature. In many countries where polygamy is legal, Muslims are known to practice it, but the practice for both faiths today is not so rooted in their theologies that one cannot be Muslim/Mormon unless one ascribes to polygamy.
As a footnote, I would add that it seems a travesty to me that many Americans still insist on Mormons being preoccupied with polygamy; it’s a travesty that many Americans discount Muslim belief because of marriage customs in foreign countries.
More could be said about the similarities/differences of these two faiths, but this will probably have to suffice, given the forum here.