Its Polygyny. But there are two married couples.
Tricky question, not knowing in what situation you are asking (per current law in the US/UK/Etc., per the law there and then, etc.), but for the sake of answering the question, I'll assume you mean 'Is it legal in his culture, at that time and place, to do so.'
The answer is yes, on the basis of anything not defined as Illegal is normally assumed to be Legal. On Luna, that is, and assuming they are correct in the doctrine they espouse that they are a separate nation (which is true by the end of the novel.) If you listen to what specific members of the Lunar Authority say, then they may be covered by any number of Earthly laws, and it becomes a much more complicated question.
As Manuel points out:
“Yes, but—Well, Luna City must have a city clerk. Perhaps you call him ‘county recorder.’ The official who keeps track of such things. Deeds and so forth.”
I said. “Don’t think so, madam. Some bookies do notary work, witnessing chops on contracts, keeping records of them. Is for people who don’t read and write and can’t keep own records. But never heard of one asked to keep record of marriage. Not saying couldn’t happen. But haven’t heard.”
However, the beginning of the next chapter really addresses the legality:
Took Stu all day to get case transferred to an F.N. court and dismissed. His lawyers asked to have it tossed out on “diplomatic immunity” but F.N. judges did not fall into trap, merely noted that alleged offenses had taken place outside jurisdiction of lower court, except alleged “inciting” concerning which they found insufficient evidence. Aren’t any F.N. laws covering marriage; can’t be—just a rule about each nation required to give “full faith and credence” to marriage customs of other member nations.
There would be no legal barrier to two couples marrying. There are extensive discussions of group marriages, co-husbands, co-wives and intermarried polygynous groups in "The Moon is a harsh Mistress"
It's exlicitly states that there...
...are no laws--except Warden's regulations--and Warden doesn't care what one Loonie does to another.
As a matter of fact, marriages aren't even legally ratified other than through the normal contract law system;
I said. "Don't think so, madam. Some bookies do notary work, witnessing chops on contracts, keeping records of them. Is for people who don't read and write and can't keep own records. But never heard of one asked to keep record of marriage. Not saying couldn't happen. But haven't heard." "How delightfully informal! Then this other rumor, about how simple it is to get a divorce on the Moon. I daresay that's true, too?" "No, madam, wouldn't say divorce is simple. Too much to untangle. Mmm... take a simple example, one lady and say she has two husbands--" "Two?" "Might have more, might have just one. Or might be complex marriage. But let's take one lady and two men as typical. She decides to divorce one. Say it's friendly, with other husband agreeing and one she is getting rid of not making fuss. Not that it would do him any good. Okay, she divorces him; he leaves. Still leaves endless things. Men might be business partners, co-husbands often are.