I’ll preface this by saying that the Old Persian word ariya used in the DNa incription, much like its Sanskrit cognate ā́rya, is a self-denominator. Whilst the word anairya (non-Aryan) exists in reference to other peoples, neither term has anything to do with race. Related to the older Avestan airya found in the sacred texts of Zoroastrianism, it essentially defines one or more groups of people with the same linguistic, cultural, and religious background we now know as Iranians or Old Iranians. In our case, Aryan and Iranian are interchangeable.Broadly speaking, the old Iranian world comprised most of Central Asia (where Zoroastrianism in all likelihood developed), modern-day Afghanistan, and modern-day Iran, where the people later known as Persians emerged. Crucially, for our discussion, the religious background is usually understood to have involved the worshipping of Ahura Mazda, along with older Iranian deities who still played a role in Iranian religious life depending on the place and period.

That being said, was the mention of an Aryan origin important? Possibly. Most, if not all, of the inscriptions Darius I ordered were aimed at emphasising and legitimising a kingship he had obtained through a coup where he killed the alleged Bardiya, son of Cyrus the Great (who this specific Bardiya actually was is another matter). The most famous example is in the Behistun inscription, where Darius lists his ancestors all the way to Achaemenes and explains that not only was Achaemenes his great-grandfather, but also the great-grandfather of Cyrus the Great, thereby inserting himself in the dynasty of the first King of Kings and legitimising his rule. The mention of an Aryan origin seems to follow a similar pattern. The line you’re referencing goes:

Vištâspahyâ puça \ Haxâmanišiya \ Pârsa \ Pârsahyâ puça \ Ariya \ Ariya ciça

“Son of Hystaspes, an Achaemenid, Persian, son of a Persian, an Aryan, of Aryan origin”.

Not only Darius, but his father (and likely Achaemenes himself) are Aryans. This mirrors another line in the Behistun inscription where Darius says the all his ancestors have been kings/royal and as such his is a royal family.

Why being of Iranian descent gave more legitimacy is uncertain.In another passage, Darius talks about the rebellions he suppressed and specifically mentions some led by Iranians (Persians, Medes, Parthians, and Sakas, amongst others), emphasising that being Iranian gave right to rule over other Iranians. A possible explanation is that, according to royal ideology, the right to rule was granted by Ahura Mazda himself, whose cult, as I mentioned, was part of the Iranian identity at the time. Therefore living according to Iranian customs may have granted a person more legitimacy and/or recognition.

For what is worth, a similar situation is found in the Rgveda, where ā́rya is essentially anyone who speaks the ā́rya language (in this case, Vedic Sanskrit) and practices the ā́rya religion – reflected in the fact that, of all the ā́rya kings mentioned in the Rgveda, 30 or so of them have non-Indo-European names and, as such, have been assimilated into the culture.

As I said, though, it’s uncertain. Of all the occurences of the word ariya, these are the only ones where it’s used in connection to kingship, and they’re all in the same context, i.e., the wars fought by Darius after his coup. The only other occurence, other than in personal names such as Ariaramnes, is in the Daiva Inscription that tells of a rebellion suppressed by Xerxes I in the Eastern territories of the empire. The people living there are referred to as ariya (Iranians) and are accused of worshipping daiva (false/evil gods) instead of Ahura Mazda. Since we don’t have similar accusations for non-Iranians in the empire (such as the Babylonians or the Phoenicians), it reinforces the idea that religion was an important part of the Iranian identity.