memes

No, it isn't the same. Islam functions differently than Christianity there. The understanding of 'being Muslim' often transcends theology and encompasses culture, history, and family lineage, rather than just what you do on the weekend. I can easily see how a Muslim person who isn't religious would still call themselves Muslim even if they practice the same amount of religion as a lapsed Christian who no longer identifies that way. Identity isn't monolithic there either despite being 90+% Muslim. It is a multi-ethnic state. While the majority are Shia, you have Kurds (who are largely Sunni Shafi'i or Alevi/Yarsan), Baluchis and Turkmen (who are Sunni Hanafi), and Arabs (who are ethnically distinct but largely Shia). Asking "Are you Muslim?" misses the complexity of it all.

The US is a settler colony that was founded on a concept of separation of church and state as described by Jefferson in a letter to the Danbury Baptists and the constitution of the US states there will be no established religion or religious tests for office this creates a "wall of separation" between religion and government. Additionally many non religious people, Muslims, Jewish people, Non-Trinitarian Christians like Mormons and other belief structures have been present since the beginning. This plurality of belief creates a self reinforcing secular culture and civil religion. Americans have a civic identity (the Constitution, the Flag, "Founding Fathers" like "Saint" Jefferson) that creates a unified identity separate from faith. This allows for that "wall of separation" where one can be American without being Christian. In Iran, the state currently fuses religious and national identity so to them being Iranian is tied to being Muslim so people in this current environment would say they're Muslim when asked because saying no would isolate them from the entire society. There are cracks there though, Persian culture is ancient and distinct, containing deep pre-Islamic traditions (like Nowruz) that some hardliners might consider haram. Because of this, there is a tension between National identity (Persian) and Religious identity (Muslim). If forced to choose, many Iranians in my opinion may prioritize their heritage over their religious label. This is complicated by being under siege by the west though.

Iran is an explicit theocracy where people are defined as muslims by default and where irreligion/atheism is not a valid category in identification by the government.

Sure, but almost all of Mexico identifies as Catholic but church going is mostly reserved for special occasions like baptisms and the big religious holidays. You'll see the crucifixes, the Jesus pictures and whatnot but it's not like they're on that annoying protestant bullshit.