Kepler, as far as I can make out, there was no Median Empire. Cyrus defeated Astyages, the last Median king ca. 550 BCE. He then merged Media and Persia into the Medo-Persian Empire, before attacking Babylon in 539 BCE. Darius the Mede “received the kingdom” from Cyrus the Persian (Dan. 5:30). This is confirmed by Dan. 9:1, which says Darius “had been made king over the kingdom of the Chaldeans,” courtesy of Cyrus the Persian. Darius was standing in for Cyrus, while Cyrus was involved with his military campaigns. Thus, even a short-lived “Median Empire” cannot compare to the Babylonian Empire (612 – 539), the Persian Empire (539 – 332), or the Greek Empire (332 – 165 at least).
Darius as title: In the Biblical record, the name is applied to three kings, one a Mede, the other two Persians. In Greek-English Lexicon by Liddell and Scott (pp. 370A, 691A) Greek form Dareios is related to Old Persian Dārayava(h)uš ‘upholder of the Good,’ or ‘maintaining what is good,’ being a Greek form of Persian darâ, meaning ‘a king.’ According to Herodotus 6.98 = Gr. ἐρξἰης or ἐρξεἰης, i.e., Herxieis, which means “one who restrains” (Latin: coercitor) or “the worker/doer.” Thus, it may be possible that “Darius,” in the case of Darius the Mede, may have been used as a title or throne name.
Some assert Ugbaru is Darius the Mede: According to the so-called Nabonidus Chronicle (cf. ANET, pp. 305ff),[1] “[In the seventeenth year (of Nabonidus)].… In the month of Tishri, when Cyrus fought at Opis on the Tigris against the army of Akkad, the people of Akkad revolted.… On the fourteenth day, Sippar was captured without battle. Nabonidus fled. On the sixteenth day, Ugbaru (Greek, Gobryas), the governor of Gutium, and the troops of Cyrus entered Babylon without battle. Afterwards, when Nabonidus returned, he was arrested in Babylon.… In the month of Marchesvan, on the third day, Cyrus entered Babylon.…”[2]
Where does Belshazzar fit in: Belshazzar is referred to as “king” in Daniel 5:1–30. Cuneiform temple receipts from Sippar attest that Belshazzar presented sheep and oxen there as “an offering of the king.”[3] While it is true that no cuneiform record refers to Belshazzar by the explicit term sharru (“king”), it is clear that during the latter years of Nabonidus’s reign, while the latter made his headquarters at Teima in Arabia, Belshazzar ruled as his viceroy, with all the authority of the king. That this fact was well known to the author of Daniel is clearly implied by the fact that in Daniel 5:7, 16 the viceroy could promise to the successful interpreter of the handwriting on the wall only the honor of third ruler in the kingdom. Obviously Belshazzar himself was only the second ruler.
[1] ANET Pritchard, J. B., ed. Ancient Near Eastern Texts Relating to the Old Testament, 2d ed. Princeton University Press, 1955.
[2] Hartman, L. F., & Di Lella, A. A. (2008). The Book of Daniel: a new translation with notes and commentary on chapters 1-9 (Vol. 23, p. 191). New Haven; London: Yale University Press.
[3] R. P. Dougherty, Nabonidus and Belshazzar, (New Haven: Yale University Press, 1929), p. 88.