There is an important difference between Islam and Christianity, which is that for various reasons Christianity developed into acceptance of a de facto separation of church and state very early on. So for example, priests and other "clerics" would be tried for crimes in separate Church courts, but the Church conceded that laws and judicial process for the rest of citizens were the province of the state. This is party for historical reasons and partly because of Christian scriptures. Jesus's words "render unto Caesar the things that are Caesar's and to God the things that are God's" were taken to justify this separation, plus the idea that the new testament largely superseded the old meant that the detailed rules and laws in Leviticus about how people should behave could be seen as no longer relevant.
Islam is different from this. From the beginning, Muhammed was a secular ruler, and there is a lot in the Qu'ran and hadiths about what we would consider to be "secular" matters to do with law, punishment and how society should be ruled. The concept of separation of church and state is essentially hostile to the core Islamic scriptures. While it is (fortunately) true that Muslims are told in their holy texts to obey the laws of non-Islamic states if they are living there, there is still an emphasis that this is a matter of expediency, and not how things ought to be.
In this sense, fundamentalist text-focussed versions of Islam are not "just" a religion, but also a political ideology. And a very controlling political ideology, similar to something like say Maoism in the detail of how people are expected to live their lives, how they should be ruled, and what the consequences are of deviating from "the party line". Which is one reason why its core texts and beliefs should be open to criticism, in the same way we would criticise other political ideologies.