At last, an r/AskHistorians thread I feel qualified to (partially) answer!

I'm not a historian, but a music major. While it is true that much of our modern concept of the musical modes comes from the Catholic Church, this is not, as you rightly pointed out, the origin of many of the terms associated with the individual modes.

Dividing musical tonal sequences into various groups that are thought of as being particularly effective at invoking certain feelings and emotions in the listener goes back as long as historical accounts exist. The oldest one I am aware of (and the one most relevant to our conversation) is from Plato's Republic.

In Plato's Republic, the wise Socrates discusses with Plato and Glaucon (Plato's older brother and a musician) all aspects of what makes a morally sound Greek, and one of these aspects is indeed music. From this conversation we learn that:

"The mixed Lydian... and the tense or higher Lydian, and similar modes.. these, then,” said I, “we must do away with... for they are useless even to women who are to make the best of themselves, let alone to men.” p 398

Furthermore, drunkenness and laziness are seen as being promoted by 'soft' modes like Ionian and Lydian

“Will you make any use of them for warriors?”
“None at all," [Glaucon] said.

So varying forms of Lydian which are described as "dirge-like" and seen as useless to creating an energetic and useful member of society, and we have Ionian and Lydian which are described by the Greek musician Glaucon as being soft and dainty, absolutely unfit for the listening of true Greek warriors! This entire ancient concept of a musical hierarchy is very reminiscent of how we look at various styles of music today. Some rock doesn't go 'hard' enough, some rap is seen as too 'fake.' What we have in the Republic is just that, a select group of individuals essentially discussing their favorite genres.

What then, according to Glaucon and Plato, is a good mode of music?

"...it would seem that you have left the Dorian and the Phrygian.”
“...That mode ...would fittingly imitate the utterances and the accents of a brave man who is engaged in warfare or in any enforced business, and who, when he has failed, either meeting wounds or death or having fallen into some other mishap, in all these conditions confronts fortune with steadfast endurance and repels her strokes. And another for such a man engaged in works of peace, not enforced but voluntary, either trying to persuade somebody of something and imploring him—whether it be a god, through prayer, or a man, by teaching and admonition—or contrariwise yielding himself to another who petitioning or teaching him or trying to change his opinions, and in consequence faring according to his wish, and not bearing himself arrogantly, but in all this acting modestly and moderately and acquiescing in the outcome. Leave us these two modes—the forced and the voluntary—that will best imitate the utterances of men failing or succeeding, the temperate, the brave—leave us these.”

So it would seem the Dorian and Phrygian modes lend themselves best to creating an atmosphere of great work ethic, reverence for authority, and perseverance, all traits valued greatly in Ancient Greek society.

It is important to remember that these ancient modes are not the same as their modern counterparts, they simply share a name. We have no way to know what tonal sequences actually made up the scales being used by the likes of Glaucon, but the names have survived through the ages, appearing in the likes of John Milton's L'Allegro:

Or sweetest Shakespeare, Fancy's child,
Warble his native wood-notes wild.
And ever, against eating cares,
Lap me in soft Lydian airs

and his most famous work, Paradise Lost

Appear'd, and serried Shields in thick array
Of depth immeasurable: Anon they move
In perfect Phalanx to the Dorian mood
Of Flutes and soft Recorders; such as rais'd
To hight of noblest temper Hero's old
Arming to Battel, and in stead of rage

TLDR: While our modern musical modes share many of the same names as modes of the ancient Greeks, the mode they would call 'Dorian' and that we would call 'Dorian' are two different tonal sequences, as we have no actual way of knowing ancient Greek music theory.

I will let a more knowledgeable historian discuss how we handed down these names through the ages so that we still call our modern modes by these ancient names, but it is worth noting that no single mode hails from a specific culture as your question would imply. The Greek Ionians were not (to my knowledge) famous for playing our modern Ionian mode (the major scale), they simply share a name.

I look forward to more replies to this thread, as I would love to learn more about the history of the modes myself! This is my first r/AskHistorians submission, so please be gentle dear moderators.

Source:

Plato. Plato in Twelve Volumes, Vols. 5 & 6 translated by Paul
Shorey. Cambridge, MA, Harvard University Press; London, William
Heinemann Ltd. 1969.