I understand, at least from Wikipedia, that he's referencing gossip and scandals at the time. But a lot of the song lyrics seem to be suggesting society was now prone to scandal or somehow salacious conduct was more acceptable generally. Was this a commonly held opinion at the time? Was it really "anything goes" in the 30s? Especially since I understood the 20s to be considered more "open" than the following decades...
The show is dealing with a lot of satire of the time. You've already mentioned the 20s as being more "open", and in some cases Porter is commenting on that and a continuation of that. We saw instances of gender norms being bent, with certain female actresses appearing in menswear. Katherine Hepburn had just come on the scene. We saw hints in popular media at homosexuality here and there. In terms of Hollywood, the envelope had been pushed to the point where the Hays Office created formal guidelines for movies. This began in 1934, the year Anything Goes debuted. The early days of Hollywood are called "pre-code", and in those days cities across the country had their own censorship boards for films. The 30s saw the rise of Mae West and her double entendres, and foreign films like Ecstasse, which featured scenes of nudity and female orgasm (not hardcore, but enough to trouble censors). This was the era of the "It" Girl. There are also a lot of salacious exploitation films or other gilms that titilate ipfren disguised as documentaries in some way. There were "nudist colony" movies (sorry, I can't think of one right now), and films like Tabu (1932) which showed nude breasts on native women.
Traditional institutions like marriage were being undermined in popular entertainment. For example, the successful stage musical The Gay Divorce, also a Cole Porter show. When the film version ultimately canme in 1934, the new Hays Office made them change the title to The Gay Divorcee so as not to encourage the notion that divorce itself could be good.
Anyway, I'm sure others are more knowledgeable than I about many other societal aspects of cultural upheaval. But I want to mention also the character of Reno Sweeney is clearly based on Aimee Semple McPherson, a controversial revivalist evangelist woman of the time ( her heyday was the '20s). She founded the Foursquare Gospel Church and would have big theatrical meetings. Frank Capra's film The Miracle Woman is also inspired by her. She's a fascinating figure in American history and the rise of Pentacostalism. In a way, she probably represents upheaval in the religious world.
Anyway I defer to those with more knowledge, but thought I'd contribute a bit from my knowledge base.