Fun fact: "Iran" means "the land of the Aryans".
But it's worth noting that in German "Alman" has become the equivalent of "gammon" in the UK, especially among immigrants and in youth culture. I.e. a derogatory slur for a specific type of native-born person (usually men) holding naïvely reactionary views. Not necessarily intentionally racist but always ignorant.
> the name is baked into everything (modern and historical) from Hebrew to Persian to Arabic to Assyrian.
The term is also used in Han China for the Greek kingdom that came to exist near them. There would have been no real awareness of the original region of Ionia, though there was rudimentary awareness of the Roman Empire a bit later.
This suggests that the reason for the name isn't really known, but no one thinks it might refer to the Greeks.
On the back of this, though, in many countries in the Middle East the Latin names are used for most countries in Europe, including Germany, which is referred to as Alemaan (from proto-Germanic Alemanniz[2]).
[0] https://en.wikipedia.org/wiki/Ionians
[1] https://en.wikipedia.org/wiki/Ionia
[2] https://en.wiktionary.org/wiki/Alemanni