Flavor text is a board game term referring to any writing in the design element that does not impact any strategy or outcome. The most common use is in games that use cards to take actions and often have a witty quote beneath the artwork. Many gamers will ignore this, but some will read the cards in their hand, especially if other players are taking long turns.
Many picture books have text with in the illustrations and unless there is a term for it, I would call that flavor text as well. Examples would be a book titles on a shelf or company names on trucks, and puns are often used to play off the story or reference pop culture in some way (in Shark vs. Train there is a moving van that reads A. Fonzarelli Moving Co., being jumped by the shark on the motorcycle, which references the ratings-desperate Happy Days episode when the Fonz jumped a shark tank on a motorcycle, giving rise to the phrase "jumped the shark"). Most people won't take note of this "flavor text", but it might amuse some its intended audience, the adult reader. (also in Shark vs. Train there are many little word bubbles that I imagine would be written by the writer since they are a greater part of the story, so I'm not speaking of those as much)
I'm wondering how that flavor text happens. Does the writer include notes as an appendix to the manuscript? Does the editor ask the writer to add some of these jewels? Does the illustrator do that completely on their own? Do illustrators feel they need permission from the "wordsmiths" to add any words to their art?