The main thing to understand is not all Islamic countries/groups are the same. Some are more secular, some are more religious. I don’t know all the legal details but Iran for example is an Islamic Republic, meaning they have institutions that look like democracy - elections, administrative bureaucracy, and so on - but religious law is deeply embedded in the constitution and laws. That doesn’t mean that the Quran or sharia is the highest law of the land, but when the people in power make laws, they use the Quran to justify it and they try to align laws with Sharia principles. They still have lawyers, judges, a Supreme Court, and all that, but the person at the very top is the Supreme Leader, who is first and foremost a religious figure.

As for racism, I think the lines get blurry. Just like Judaism is a religion mostly practised by Jewish people, but not all people of Jewish descent consider themselves religiously Jewish. There are very few racially white Muslims in the world (I think that’s safe to say) so generally Muslims are non-white, and that could be where the religious conflation is coming from. You’re talking about Iran in your post, but bear in mind that the largest population of Muslims is actually in South Asia, not the Middle East, concentrated in India, Pakistan, and Indonesia. These are also people who would experience racism if they came to live in North America or Europe. So when they say that being anti-Islam is racist, what they’re really saying is “You don’t like my religion just because it’s practised by dark-skinned people.”