However, I disagree with your conception of Islam as a state, even if it was explained to you by Muslims. The strongest argument I can build from your statements is that, according to the reference to the end of the Sunni Caliphate in 1923,
p1) only Sunnis are Muslims, and
p2) the Caliphate is unique, and
p3) the Sunni Caliphate of 1923 is the original one, thus
c) it was the state of Islam.
We can disprove all of these premises. p1) is obvious, there are more Muslim religions than just Sunnis. The earliest schism was the Sunni-Shiites split, happening immediately after the first prophet's death.
About p2), while I'm fuzzy on the details, I'm pretty sure that between the 900s and the 1900s there were at least 3 major, parallel Caliphates and also a bunch of smaller Caliphates. Geographically they were even sometimes overlapping. It might be interesting that the Caliphate of the Ottoman Empire (the one in question) was a Hanafist (a Sunni splinter group) Caliphate.
On p3), the Sunni caliphate of 1923 was reestablished after a 300 year "hiatus" by the Ottoman Emperor to lay claim on Crimea. It had no representation besides a leader, the Sultan. Before the dissolution of the major Sunni Caliphate in the 1500s it relocated several times, from today's Syria to today's Iraq, to then and now Egypt. Thus we can say that the Caliphate had no continuous existence. We can furthermore say that the time the Sultan of the Ottoman Empire was the Caliph, it was because it was a diplomatic ploy of the secular power of the Ottoman Empire.
Therefore, c) must be wrong. There are more Muslims than Sunnis, the Sunni Caliphate wasn't unique, and the Caliphate that ended in 1923 was not the original one.
A less philosophical counter-argument could be the vigorous infighting between different Muslim groups we see today. I'm curious how the war on Iran changes that, if at all.
... every group of every monotheistic religion says and believes they're the only "true" group, their group is the only valid group, and the entirety of that religion. Islamic dogma states very clearly, and every muslim will repeat it, that there is "only one islam".
This despite the fact that what you say is correct. There's 100s, minimum, of different versions of islam.
Your idea, that history is clear proof to the contrary ... well history is clear proof that there is no god and therefore no valid religion. In the case of islam, one might point out that the central promise of islam as a religion is that muslims will win militarily, because god will intervene directly (but "of course" what is currently happening in Iran proves they are wrong and every other group of muslims is right - this is the sort of argument you're up against). The fact that any caliphate fell at all is a pretty damn obvious contradiction to the entire religion.
Frankly, I must say, I like the "goal" of Christians and Jews a whole lot better.
And, frankly, while this is most prominent with Islam, that religions describe their goal to be a single state and trying to be a single state is the norm, not the exception. Christianity is the exception here that does not want to have state power (even though that rule screams "compromise with the Roman emperor", and hasn't exactly been followed very well once Christians were well established)
So no more muslims allowed in the US then? In fact no religion allowed except Christianity or revering the US directly somehow?