For the Streisand effect to be an example of itself, there would have to be an attempt to ban or suppress the usage or knowledge of the idea of the effect. I don't think that's happened, so I don't see how it's self-referential.
"It is named after American entertainer Barbra Streisand, whose attempt to suppress the California Coastal Records Project's photograph of her residence in Malibu, California, taken to document California coastal erosion, inadvertently drew further attention to it in 2003."
My point is that more people know about Barbra Streisand and the Streisand Effect because of Streisand's attempts to suppress information.
the Streisand Effect is fake news.
Shit, now I feel old.