I never saw Obama as a muslim. I always felt he was an atheist. His religion like all progressives is his ideology.
IMO the reason progressives side with muslims is that sharia is more compatible with communism. Submission is what they are looking for.
The reason Progressives side with Muslims is because they’re both anti-Christian. I think you’re on to the liberty angle, but following the thread of history and you don’t find liberty growing up out of either philosophy or worldview. Liberty is a uniquely Christian aberration to the history of mankind.
Eliminate the nuisance of the Christian idea and ideals and you go a long way back toward totalitarianism. Furthermore, both Islam and Progressivism are intolerant, elitist, and socialistic. Statists flock together.