Do you think this is particularly pronounced amongst Muslims? I mean the whole thing with it being immutable - as if you can no more change your religion as a Muslim, than you can change your own brain, legs or eyes?
I think it is rooted in a theological idea and the cause of a lot of inflexibility that leads to different problems.
Your instincts and ideas on this are pretty much spot on. It's about asserting control through religio-ethnic identity politics. The supremacist religious ideology that is often preached to Muslims in the West is a far more compelling (and a less transparent power grab by the elite) means of control than a simple ethnic identity. But it's all a crock of shit. In the West, Arab muslims are separate from Subcontinent muslims are separate from African muslims, etc. Still, people's sense of identity becomes tied to a myth of Islamic identity and brotherhood, making apostasy seem at the very least unconscionable and at worse impossible short of brainwashing, extortion, insanity, etc.
I don't think there's any theological basis for it other than the draconian nature of the apostasy punishment. After all, the primary sources of Islam are full of accounts of the dissenting voices from Muhammad from within the same racial group/tribe.