From the title, you can tell that I don't actually know what this phenomenon is called, if it has a name at all, nor if it has already existed for a long time. My own searching has been unhelpful, for this reason.
Essentially, it's when one refers to something as "not-X", in order to draw attention to a similarity to X while not being X, usually in a derogatory manner.
Examples include:
A character going to an alien planet in a cartoon, and the aliens there have "not-cats" and "not-dogs".
After the popularization of the game "Dark Souls", many "souls-like" games were created, much in the same way an explosion of first-person-shooters appeared after Doom. These games could be (and occasionally are) referred to as "Definitely Not Dark Souls" or Not-Dark Souls the same way fps games were called "Doom-clones".
Some believe that the band The Monkees were created to capitalize on the success of The Beatles, and called them The Not-Beatles or Not-The Beatles (source, my mother).
Is this technically correct grammar? Does this have a name? And how old is this sort of phrase? I find myself saying things like this often, and my meaning is always understood, so it must not be entirely new. However, it does feel recent, as unscientific as that sounds.
For another amusing example: the google results for "Not-Mario" will return several listings for Nintendo products, including Mario branded things, as expected… but then the Wikipedia page for Luigi. Because he's Not-Mario.
Best Answer
The OED has examples going back to the late 16th century, both with hyphen and as two separate words (the examples I give here are with the hyphen). It seems to have been common in situations where we might use the prefix "non-" or other methods of negation: "For not-payment of ministeris stipendis." (1590) "The eater despised the not-eater; and the not-eater iudged the eater." (1622) "If both the Beheaded and the Not-worshippers of the Beast were of the same kind." (1680)
There aren't many 18th century examples, but we also find the Victorians at it: "What He attributes to idols, i.e. not-gods." (1860) "A complete idea of a closed figure, and of the boundary which incloses it—the outline separating object from not-object." (1867)
It's hard to be certain whether the OED's examples reflect genuine patterns of usage, but they also have several from the 1990s which might indicate a new burst of popularity (as well as changing patterns of how they collect usage examples): "It would differ from..matter because of its immobility. It would be almost a not-being." (1995) "If he's including himself among the not-movers and not-shakers, socially unconscious, and poorly educated, why should anyone vote for him?" (Usenet, 1992)
You might surmise an increased popularity of "not" since the 1990s is connected with the film ''Wayne's World'' (which popularised "not" as a stand-alone negation) or computing, both of which the OED mentions. But it is an old way of forming a negative, which has persisted on and off for over 400 years.
Reference: "not, adv., n., and int." OED Online. Oxford University Press, June 2021. Web. 23 August 2021.