I imagine if any European medievalists were to come across this thread they might point that there is an enormous amount of debate about whether Europe even had a "feudal structure" to which the Muslim world could be similar.

I'm not able to address that debate, but suffice it to say that it really depends on what you mean by feudalism. And what you mean by feudalism, depending on how expansive your definition is, might not have ever existed in Europe, or might only have existed in a limited fashion in certain corners of France and England.

That being said, taking a sort of maximally defined understanding of feudalism, we might ask something like "Did the Muslim world have hierarchical structures based on land holdings that generally focused on the self-preservation of military elites?"

The basic answer is: Yes. In the Islamic world, the most frequently compared system to "feudalism" is the iqta. The iqta was basically a form of tax farming: The Sultan doesn't have the time, energy, resources, or technology to manage all of his lands efficiently, so he gives parcels of lands to someone (frequently but by no means always a military commander) for them to control the land in exchange for a percent of the tax profits.

But the features of the iqta system were very different from the features of classical descriptions of feudal systems. For one thing, there were different types of iqta. While the person granted the iqta might manage the land directly and then pay the tax upwards, there were other iqta grants where the Sultan collected the tax himself and then simply paid the money downwards.

Moreover, where the grants in a feudal fiefdom entitled the noble receiving the land to own it in perpetuity (even the words we use to describe this relate to these systems: "entitle"), in the Medieval Islamic world the Sultan could revoke any iqta at any time at his pleasure. Now, in practice, many of these grants ended up being hereditary, but there was no equivalent of a class of landed nobility in the Islamic world to match, say, the barons who launched the Barons' Wars in England.

There were other important differences from traditional descriptions of European feudal structures in the medieval Islamic world. For one thing there was no concept of serfdom in which the peasantry were tied to the land in perpetuity. Medieval Islamdom also had a very interesting relationship with slavery, in that in many regimes it was slaves (who often spoke a foreign language like Turkish) who were the elite military class that formed the rough equivalent of European nobility. This also often placed a greater emphasis on native-language-speaking locals who were of no consequential family background gaining significant power in forming the clerical, legal, scholarly, and religious learned classes that mediated between the often foreign Sultans and the local population.

This mediation system has been called the "Amir-Ayan" system, two Arabic words that might be translated as something like the "Military Commanders and Local Notables" system. I'm not familiar with any equivalent of this system (which, again, was not actually a uniform or designed system by any means) in medieval Europe, but it was certainly a kind of hierarchical system designed to enable governance.

I could go on with some other examples, but for some very good and actually quite short further reading on this and how medieval society worked in an Islamic context I would recommend David Morgan's Medieval Persia 1040-1797, or for a much longer introduction Marshall Hodgson's The Venture of Islam, particularly volume 2.