The two rivers have had many names over the past 8,000 years. Many of those have been lost to history. The "Caucasians" who first inhabited the area obviously named these rivers. Sumerians (non-Semites) had names. The Proto-Semites from Arabia gave names as did follow-up migrations of Arabians to the Area (Amorites, Akkadians, Assyrians, Aramaeans, and Babylonians gave Semitic names (which I do not know.)
Indo-European Kassites, Mitanni, Medes, and Persians (Achaemenid) later gave some names for the rivers.
Alexander the Great, a Greek Speaking Macedonian conquered the vast Persian Empire that included Mesopotamia. The Ancient Greek name Tigris was borrowed from Old Persian Tigra-, which was in turn borrowed from Elamite ti-ig-ra, itself from Sumerian idigna. Euphrates is Greek for “sweet water.” The long time under the Seleucid Greek Speaking Empire held the area until the rise of the Indo-Iranian Parthians. Later it passed to the renewed Persian Sassanid Empire. Apparently they kept the Greek names for the rivers in public usage. Arabs later conquered the Area from the Sassanids and renamed the rivers with Middle Arabic names.
The Arabic name, which I do not recall, was used for the section of water where the rivers converge before emptying into the Persian Gulf. During the Ice Age, this confluent river ran hundreds of miles into the almost empty Persian Gulf (a vast valley), until the rising sea levels breached the land bridge at Bandar el Abbas (where Arabia almost touches Iran). The sea level rise sometime between 11,000 and 9,000 the sea poured across that land bridge flooding the Persian Valley and the long river of the Tigris-Euphrates convergence and its fertile valley became new seafloor for the Persian Gulf.
Amergin