The Odyssey absolutely and explicitly sets up a double standard, but it's not the one OP's question claims.

In book 5, we're introduced to Our Mighty Hero...sitting on a rock on an island beach, staring at the sea and sobbing. He is somewhere been stranded and an actual prisoner on the island belonging to the "lovely goddess" Calypso. The poem implies that there was some mutual attraction between them initially. But in the years he's been on the island, that has evaporated. They sleep together every night, sure, but Odysseus isn't happy about it. It is possible, although not definitive, to read the situation as coercion/rape.

But this isn't the concern of the poem itself, which doesn't frame this as a "normal" relationship between men and women. Why not? Because Calypso isn't a woman, she's a goddess. When Hermes comes along to pass on the order for her to help Odysseus leave the island, that is the framework with which she protests. From the Penguin Classics translation:

You are hard-hearted, you gods, and unmatched for jealousy. You are outraged if a goddess sleeps openly with a man even if she has chosen him as her husband. You were the same when rosy-fingered Dawn fell in love with Orion. Free and easy yourselves, you were outraged at her conduct, and in the end chaste Artemis of the golden throne rose, attacked him in Ortygie with her gentle arrows and left him dead. And so again, when the lovely Demeter gave way to her desire and made love with her beloved lasion in the field of the three ploughed furrows, Zeus heard of it quickly enough and struck him dead with his blinding thunderbolt.

The contrast with, of course, Zeus is clear. Calypso gets to essentially denounce all of Greek mythology as sexist, even if it is not a sparklingly feminist message today.

"Adultery" in the traditional sense, thus, is not the issue in this case.

That said, K.J. Dover and other scholars are exquisitely clear that under Athenian law (to be fair, not necessarily the situation reflected by tradition of singing the Odyssey), adultery with citizen women was very not okay for men. Naturally, it was a crime committed not against morality or the polity, but against a specific man who had legal control of the woman involved:

But Greek laws were not lenient towards adultery, and moikbeia, for whch we have no suitable translation except "adultery," denoted not only the seduction of another man’s wife, but also the seduction of h s widowed mother, unmarried daughter, sister, niece, or any other woman whose legal guardan he was.

Dover adds that adultery was considered worse than rape, because rape was considered a crime of passion and the moment whereas to carry on an affair involved seduction--that is, a long period of time and effort. Xenophon's Hiero, which is more or less a work of political philosophy, compares the two:

At any rate, it is not uncommon for the laws of communities to allow people to kill seducers, and only seducers, with impunity, and the thinking behind this law is obviously that seduction impairs the affection a wife feels for her husband. After all, if sex takes place without the woman’s consent, this does not make the slightest difference to the regard her husband feels for her, as long as the affection she feels for him remains inviolate.

Now, it's important to keep in mind that we're dealing primarily with normative or prescriptive sources here, not descriptive. (And Xenophon, for his part, is exaggerating a bit for rhetorical effect, as Edward Harris points out--seduction/adultery was by no means the only crime for which one could be executed.) The cuckolded man was afforded, in theory, a significant amount of say over how/whether the philandering man was punished. So even under the law, there would be a lot of room for flexibility or ambiguity in what actually happened.

However, more literary/normative sources are particularly useful in reflecting one aspect of the situation: being seen as an adulterer was not good. The fama for such men would swirl with scorn and satire. Dover suggests that because women were seen as easily seduced and swayed by temptation meant that adulterers rather than the cheated-upon received the brunt of social derision.

But just as we have to consider the context of Odysseus sleeping with (under ambiguous terms of consent) a goddess, the status of women in relationships outside marriage also mattered. For example, enslaved women were subject to rape by their owners and by whomever their owners gave them to as a "gift."

To discuss your monogamy question, let me start by saying that it is not strictly possible to discuss Ancient Greece as a unified concept because we're actually talking about different city-states, or poleis as they were called in Greek, that developed in fairly close proximity to one another and often interacted with each other. When most people think of Ancient Greece, they are thinking about Classical Athens or maybe Sparta, but the people who lived in those different cities did not think of each other as compatriots, but as members of a different "country" altogether. The reason for this societal development is varied and interesting, but not relevant to this answer so I'll just leave it at that. Also, much of what is known about Ancient Greece actually comes to us through the writings of a very elite social class in one particular polis, Athens, so it's difficult to say that all poleis did this or that (especially given the fact that the Athenians were pretty much always convinced that they did everything the best way).

With that disclaimer out of the way, I'll say that the Greeks were legally monogamous. To perpetuate the polis, citizens were expected to birth and rear more citizens, which means they had to get married. The Greeks would marry only one woman at a time (monos (one) and gamos (marriage)), and only her children by him would be his legitimate heirs (though it was not unheard of for some men to legitimize children from other relationships). Marriage was mostly a legal and economic matter with the father of the bride turning his power over his daughter over to her new husband, who was expected to take care of her physical wellbeing and give her rule over their household, [e.g. nomos (rule) of the oikos household, where English derived the term economy], which gave her a legitimate place in the polis.

Men were, however, societally polygynous (poly (many) and gyne (woman)), meaning they were free to seek sexual relationships with other women. Odysseus' longterm cohabitation with both Circe and Calypso was certainly outside the norm of Mycenaean society and was pretty much not done by the time of the Classical historians. It is interesting to note that in the Homeric world, both these women were goddesses, which largely explains the strange nature of the relationship. His sex with Hecuba would more normally be referred to as rape in modern terms, as she was a spoil of war and therefore a slave, but it was very common for Greek men to have sex with their slaves, and there was no legal or societal sanction for these "relationships".

Source: primarily Barry B Powell's Classical Myth and Walter Scheidel's "Monogamy and Polygyny in Greece, Rome, and World History"

There are caveats to any answer; the biggest is that ancient Greece was less one society than an agglomeration of hundreds of mini societies which often differed from each other. So for almost any "rule" you can find times and places where it didn't apply.

That said, a pretty common rule overall was that sexual morality for a Greek man depended less on what you did than Judeo-Christian sexual morality, and depended more on who you did it with.

So there is no rule that states a married man shouldn't have any sex outside marriage, but there were a lot of rules about who it was and wasn't okay to have sex outside of marriage with based on the status of the people involved.

If you wanted to very broadly sum up the rules on who was and who wasn't sexually available, you could do worse than saying that people under another citizen's or a community's protection were off limits for a Greek man, and it was open season on people not such.

For instance, in classic Athens, sex outside marriage with a woman of the citizen class was a Very Bad Thing (married or unmarried their chastity was both a household and a community concern). But there was a large class of people not counted with the respectable people, not part of the honorable part of the community. These might include slaves, low-class metics (resident aliens) and other noncitizens (though not all metics/noncitizens), prostitutes or other disreputable professions (and citizens often weren't allowed in disreputable professions), and so on. You weren't impinging on the honor of anyone if you slept with any of these people, because they didn't have honor to begin with, so it was okay (unless they were attached to someone with honor, in which case his honor came into it).

For an example of this aegis, the slave women in Odysseus's house should not have been at outsiders' disposal because they were attached to the honor of Odysseus/Telemachus, and disrespecting this attracts a lot of bad commentary and eventually punishment for the suitors and for the slave women they seduce and/or rape (which one isn't stated, and it's not clear that it is relevant from the poem's POV, since either way it's about Odysseus redressing violations of his household). Zeus tends to do similarly for the Olympian household, as Calypso complains.

The bottom line is that it was OK for a Greek man to sleep around outside of marriage, but he was supposed to limit his partners to those not under the aegis of another man, household, or community that he was bound to respect.

META question (are they allowed?): Would a (ancient literature) podcast from a person with Ph.D. in literature qualify as a valid source for an answer to this question?

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