Merriam Webster defines these words as follows:
Definition of du jour
1: made for a particular day —used of an item not specified on the regular menu
soup du jour
2: popular, fashionable, or prominent at a particular time
the buzzword du jour
1a(1): occurring in or existing at the present time
the current crisis
(2): presently elapsing
the current year
(3): most recent
the magazine's current issue
the current survey
archaic: RUNNING, FLOWING
2: generally accepted, used, practiced, or prevalent at the moment
current ideas about education
I notice a pattern between sense 2 of du jour ("popular, fashionable, or prominent at a particular time") and sense 2 of current ("generally accepted, used, practiced, or prevalent at the moment"), but I don't really have the tools or the intellect to truly understand what that similarity is (the only similarity I've figured out thus far: they both share a sense of short-lived popularity). Is there really a similarity between these two words (the pattern between sense 2 of du jour and sense 2 of current)? If there is no such similarity between the words and their senses, then what are the differences?
reptaxI notice a pattern between sense 2 of du jour ... and sense 2 of current
du jour is French, borrowed into English, for "of the day", and its original, literal meaning is "of this specific day", "of today".
The connotation of "day" that is used in both of the definitions you refer to is not one specific day, but a series of days centered on today: "nowadays", "in our time", "these days", "at the present time", "as we live now".
That's the concept that joins those definitions. Time. Note the words "at a particular time" in the first of those two definitions, and the words "at the moment" in the second.
"current" is the higher register and more standard word of the two. "du jour" is perhaps more used in popular magazine style; it gives more the impression that the writer and reader recognize each other as trendy, in-the-know, with-it, hip.
"current" occurs before its noun; "du jour" occurs after.
Facebook is the current favorite among social media platforms.
Facebook is the darling du jour among social media platforms.
reptaxI don't really have the tools or the intellect to truly understand what that similarity is
I doubt it. Maybe you're looking for something more complicated than actually exists in these words. Or you're confusing difference in word meanings with difference in the environments in which words are used.