Meaning of "Almost every kinky kind was being had and enjoyed."

I saw this sentence in NYT OP:

"Almost every kinky kind was being had and enjoyed."

What does the part in bold mean? What grammar rules pertain to it?

Solution 1:

The bolded verb phrase is a passive past progressive, which is not a tense that you see every day. If we reverse the passive and restore some elided words we get the following:

[People] were having almost every kinky kind [of sex] and enjoying it.

Note that deploying the passive in this way is a kind of play on words. The idiom to have sex normally is not cast in the passive, because sex is not really a normal object of the verb have, but is rather an integral part of the idiomatic expression have sex. The statement Sex was had is unusual and surprising, though still intelligible. In the linked article, Savage uses this unusual passive construction in order to put the focus of the sentence of kinky kind [of sex] and create a parallelism between had and enjoyed.

Solution 2:

More context from your link reveals what's missing:

Not that heeding our desires always simplifies matters. One recent writer to Savage Love thought he would enjoy seeing his wife fool around with another man, and initially did: “Almost every kinky kind was being had and enjoyed.” But when his wife had vaginal intercourse with the other man, something happened.

What's missing is "sex"; the line omits that because it's supposed to be understood from context as meaning:

"Almost every kinky kind [of sex] was being had and enjoyed."

The "was being had" means the couple mentioned had been having sex.