"More [x] than [y]" is an idiomatic way of saying that something more closely resembles one thing than another. It doesn't mean that the subject is not [x], just that it should perhaps be considered to be closer to [y].
For example, if someone said they were cooking a stew, but the consistency of it was more liquid than solid, you might say "it's more of a soup than a stew". You're not saying that it isn't a stew, or that it is a soup - just that it has perhaps turned out more like a soup than intended.
The text you quote comes from a piece about the definition of persiflage, which means "light banter". The two statements you are asking about are made to try and help define what that means.
A "flapper" is an old-fashioned word (possibly 1920s) for a party-goer. A "sage" is a wise person. I believe "more flapper than sage" is saying that "persiflage" or "light banter" means talking about trivial matters, not serious matters.
Likewise, a "purse" is something carried for fashion, a "flag" is something carried to represent national politics. Again, I believe "more purse than flag" means that light banter is more likely to be about trivial things like fashion than about serious matters like politics.
I can't be 100% certain these are the intended definitions and comparisons of those 4 words, but I do feel confident that this is the intended meaning in the context of the article.