Since the French Revolution, Christian Monarchies have been overthrown by people who were also opposed to Christianity, and the new republics associated with secularism. Muslim monarchies, on the other hand, with the exception of Albania's (the only Muslim monarchy of modern times located entirely in Europe), have generally been overthrown by practicing Muslims who would never dare to attack Islam, at least not to the degree that revolutionaries in historically Christian lands have attacked Christianity, and sometimes (e.g. Iran) by Islamic theocrats acting in the name of Islam. Even in Egypt and Turkey, where Islamic monarchies were replaced by relatively secular republics, few if any republicans would have wanted to be seen as anti-Islam (and both countries now seem to be moving in a more Islamic direction). This suggests that despite a long history of Islamic monarchies, Islam is intrinsically less compatible with monarchism than Christianity is.
The decline of "Altar & Throne" Christian Monarchy in Europe has coincided with the decline of European Christianity, whereas Islam appears stronger than ever in the many countries once ruled by Muslim monarchs that are now republics. Islam has significantly more power in Egypt, Iraq, Libya, Afghanistan, and Iran today under republican governments than it did under those countries' 20th-century monarchies, but European republics are all much more secular than they ever were as monarchies. Of the Muslim monarchies that survive today, it seems that if any of them were overthrown they would be more likely to be replaced by an "Islamic Republic" than by a secular one. Despite being frequently (and inaccurately) denigrated as "medieval," contemporary radical Islam has little to do with nostalgia for the past, certainly not its monarchies. It is difficult to imagine a contemporary "fundamentalist" Muslim pining for the Ottoman Empire as a traditionalist Catholic might long for the Bourbons or Habsburgs or an Orthodox reactionary for the Romanovs.
Perhaps this is related to European leftists' seemingly puzzling fondness for Islam, which is hardly their ally on issues pertaining to sex: do they sense that it is more inherently egalitarian than their ancestors' Christianity?