I distinctly remember seeing the web page where the hoax originated
years ago, but despite my best efforts I
cannot find it again. All I was able to find was
this video, presumably uploaded by the same hoaxer. Note the upload date of April 1st, 2007. At the time I didn't really think much of it, despite seeing a smattering of forum posts on sites similar to TMK from people who believed the hoax. Since the source of the gag is gone, I honestly have no idea how it's become this much of a meme. In my attempts to find the page I keep seeing people who seriously think that Thwomp's name really
is "Marty".
As for me, as a kid I definitely knew he wasn't green, but I always assumed he was behind the cage for a reason. I guess I must've figured he came out of the cage on a higher difficulty and then somehow my brain rearranged this into a "fact" after years of not playing Mario Kart 64.
Edit: Well, after a fair bit of searching I finally found
this, which appears to be a reprint of the text on the aforementioned web page and even provides a link to the hoaxer's original website:
http://www.magikoopa.com/marty/ which unfortunately is completely gone now. It says that the hoax specifically originated on a GameFAQs thread asking about the green Thwomp, which then sort of ballooned into an elaborate hoax perpetuated by three named individuals, including the aforementioned hoaxer. No idea how long ago this happened. The YouTube video is the earliest evidence I can find.