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You mention both Greeks and Romans, so I treat each separately below, identifying exactly when they switched to monogamous laws (in the case of the Romans) as you ask:
Marriage Under the Greeks
Greek society was always monogamous. For example, in the Odyssey, one of the oldest Greek works which was originally transmitted orally, Odysseus has only one wife even though he is a great lord. In the Oeconomicus, it says
My belief is that a good wife, being as she is the partner in a common estate, must needs be her husband's counterpoise and counterpart for good...
The wife is described as the partner and implies there must needs be one. In Plutarch's life of Alcibiades a typical story of a wife is told:
Alcibiades went to the house of Hipponicus, knocked at his door, and on being shown into his presence, laid off the cloak he wore and bade Hipponicus scourge and chastise him as he would. But Hipponicus put away his wrath and forgave him, and afterwards gave him his daughter Hipparete to wife. Some say, however, that it was not Hipponicus, but Callias, his son, who gave Hipparete to Alcibiades, with a dowry of ten talents; and that afterwards, when she became a mother, Alcibiades exacted other ten talents besides, on the plea that this was the agreement, should children be born.
Unfortunately there are very few written laws known from ancient Greece, except as they are mentioned in plays and such. Greek law differed from nation to nation and there are no inscribed monuments of laws as we have for Rome. The Greeks preferred judges who interpreted traditional laws over hard and fast written laws. For this reason, there is no specific written Greek law that I know of forbidding polygamy.
Marriage Under the Romans
Under the Romans, polygamy was legal until the Christians gained control in the reign of Constantine. The first Roman law forbidding polygamy was by that emperor made in 320 AD, stating: "No married man may have a concubine during the existence of his marriage." In this law the term "concubine" refers to wife by usus, not to a slave. Usus normally required a written legal contract and the wife was a legal, freeborn wife. In Roman society a usus wife was just as legal as the primary wife. Originally, Romans had several different grades of wife and it was common for wealthy men to have all grades. This gradually devolved into having a primary wife and various usus wives. This does not include sex with slaves who were not considered any kind of wife. Roman law provided for what we now call "common law" wives, meaning automatic usus status even without a contract if the woman slept in the man's house for over one year. The Twelve Tables specifically states the terms of this law as follows:
"If a wife should sleep for three nights in a year out of her husband's house, she should not be subject to his paternal power."
Thus any woman could escape being an automatic wife by claiming she slept for three nights outside of the house. Note that any usus wife, contract or not, would be considered materfamilias (primary wife) by the law if she was the only wife of a man.
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Apsu and Tiamat were the first documented monogamous couple through the use of cuneiform. I believe they were Alalu's son and daughter. There is a lot more to them than that. However, they predate Rome, Greece, and Egypt. They originate from Mesopotamia, specifically Ancient Sumeria.
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It was probably always the norm, at least in a way that also tolerated concubines. The Ancient Greeks were of course descended from Proto-Indo-Europeans. As early as 1864, French historian Numa Denis Fustel de Coulanges reasoned in his magnus opus, La CitΓ© antique, that marriages were monogamous from the earliest days of the Indo-European peoples.
The institution of sacred marriage must be as old in the Indo-European race as the domestic religion ... The marriage ceremony, too, was so solemn, and produced effects so grave, that it is not surprising that these men did not think it permitted or possible to have more than one wife in each house. Such a religion could not admit of polygamy.
- De Coulanges, Numa Denis Fustel. The Ancient City: A study of the religion, laws, and institutions of Greece and Rome. Courier Corporation, 2012.
Modern scholarship lends his arguments support. The Oxford biological anthropologist, Dr Laura Fortunato, writes that:
The phylogenetic comparative analysis of marriage strategies across societies speaking [Indo-European] languages provides evidence in support of [Proto-Indo-European] monogamy ... More generally, these reconstructions push the origin of monogamous marriage into prehistory, well beyond the earliest instances documented in the historical record. This implies that the archaeological and genetic evidence for the nuclear family in prehistoric populations may reflect a monogamous marriage strategy.
- Fortunato, Laura. "Reconstructing the History of Marriage Strategies in Indo-European-Speaking Societies: Monogamy and Polygyny." Human Biology 83.1 (2011): 87-105.
It is difficult to verify pre-historical social practices, but monogamy must have been very ancient. At the very least, monogamy was standard by the time historical records began in both Ancient Greece and Rome. While men certainly kept concubines and had sex with slaves, these were not recognised as wives nor give birth to legitimate children.
By the historical period, by contrast, [Socially Imposed Universal Monogamy] was firmly established as the only legitimate marriage system [in Greece]: polygamy was considered a barbarian custom or a mark of tyranny and monogamy was regarded as quintessentially "Greek" ... There is no sign of an early polygamous tradition in Rome.
- Scheidel, Walter. "A Peculiar Institution? GrecoβRoman Monogamy in Global Context." The History of the Family 14.3 (2009): 280-291.
Athens under Solon the Lawmaker did not exactly outlaw polygamy per se (this was probably already illegal or socially unacceptable, excepting as concubines). Rather, it established the concept of legitimacy by excluding bastard children from the legitimate family and inheritance.
[A]s a legally sanctioned reproductive institution, the laws redefined the conjugal family as the sole legitimate family form ... after the time of Solon's laws the bastard was not considered a full member of the father's household, if he or she was a member at all.
- Lape, Susan. "Solon and the Institution of the" Democratic" Family Form." Classical Journal (2002): 117-139.