A transliteration of a very ancient word (which got passed down to the Hebrew through centuries of oral tradition, most likely from a more ancient culture with a different language) can indeed eventually become a loan word (if used regularly within the Hebrew culture.) I’ll try to explain with an example: the word Eden is thought by some to have an origin in the Ugaritic word for “a well-watered, verdant place”. So it probably has very ancient Semitic language family origins. Yet, because the Bible has had such a profound impact on English and many other languages, we now use words like “Eden” and “Edenic” to refer to an exceptionally fertile and pleasant landscape. So it has become a “loan word” by popular uses beyond its original application in Genesis.
I mentioned it because I wanted to emphasize that we can’t automatically assume that the Euphrates and Tigris rivers in the Genesis description of the Eden region’s rivers are necessarily the same rivers we assume today when we hear those names. This situation can be compared to a place name like Portland, which originally referred to a place in England but later got applied to a city in Maine and eventually to what became the largest city in Oregon.
To explain how a word can “evolve” from in its pronunciation in ancient Persian to a particular Hebrew transliteration to a particular Greek transliteration would involve a great many technical aspects of each of those languages (such as how prefixes and suffixes are applied, as well as the available phonemes of the language.) That said, I’m not sure how doing so—even if all of those ancient transformations could be explained— would be helpful.
I could similarly explain the technicalities of how the Hebrew name YEHOSHU’A became IESOUS in Greek but in English we say either Joshua, (if we transliterate from the Hebrew source) or Jesus (if we transliterate from the Greek source.) Either way it is really the same name, linking back to the Old Testament hero, Joshua. Yet, even many Christians today don’t realize that Jesus was basically “named after” Moses’ successor, Joshua. Some people reading the Bible today think of Joshua when they see the name Jesus, and some do not. That comes with the many complications of words “evolving” as they pass through various languages. By the way, the Greek form IESOUS/Jesus had to lose the SH sound from the name Joshua because Greek has no such SH phoneme. For some readers, that will bring to mind the SHIBBOLETH/SIBBOLETH story from Judges 12.
To the ancient Hebrews, F’RAT would indeed be a name that they would recognize. Whether they unambiguously associated that river with a particular place is harder to determine. Some Americans today in the Pacific Northwest know of no other city by the name of Portland than the one on the Columbia River in Oregon. Yet, many of their neighbors are aware of a Portland in Maine, while even fewer know that there is a Portland in England on which both city names are based.
There are two major rivers in the USA called Red River. Nevertheless, millions of people around the world, including many English speakers unfamiliar with those American rivers of that name, immediately associate the Red River with a waterway which begins in China and flows through Vietnam to the Gulf of Tonkin. Place names are not always unique. That is one of my major points.
Many people groups of the world have used names like Big River for the largest stream in their area. (Another example would be Fast-Flowing River or Green River.) Some of those cultures have ancient stories referring to Big River which probably came from other cultures or even from their own ancestors who lived in some distant region before a major migration. When those cultures retell such oral traditions and mention Big River, some or all of the people may even misunderstand the geography. It is very possible that the same thing could have happened with references to the Euphrates and Tigris rivers in the Genesis account. We don’t know how ancient was the oral story of Eden’s garden that the Hebrews wrote down in their own language. We don’t know how many languages that oral story may have passed through on the way to the Hebrew compilers of Genesis. We don’t know if the ancient Hebrews always assumed that the Euphrates River of Genesis was in Mesoptamia or if some had also associated the name with other regions. History in general and historical linguistics for certain can be messy that way.
The fluidity of place names brings to mind what I discovered in studying the migrations of some German Brethren families I researched. Every few generations they moved on to new frontier areas of America, and each time they applied the name Goshen to their small village. The name had its original source in the Land of Goshen in ancient Egypt, a Hebrew place name where the children of Israel had once pleasantly resided. Other pioneer groups similarly applied Goshen to their New World communities. Today I think there’s over a dozen states with cities and towns named Goshen. The same thing happened in many other countries, such as Canada, South Africa, and Australia.