My understanding is that the description of the game is a hoax, but the game itself is not.
I will quote some excerpts from the linked document that suggest it is a hoax:
The precise rules are far too numerous to list here, and the above rules merely introduce some of the unique aspects of SR Chess.
Even the page linked to does not describe the full rules of the game.
A good grasp of the more comprehensive laws that govern legal and winning patterns and sequences is essential for expert play, but these are amply documented and explained in Samuel Worthington's fourth edition of Stanley Random Chess: The Official Player's Guide - Vol. 1, The Rules (Vol. 2, The Players and Vol. 3, Developing Winning Strategy are also worthwhile).
A Google search for this book did not turn up any links to it. It apparently does not exist. All that turned up were the page linked here and copies of it.
Over 535 such variations have been documented by the ISRCA, and the appendix of their 2004 Official Stanley Random Chess Handbook summarizes the 32 more popular international variations.
When I searched Google for "Official Stanley Random Chess Handbook", I did not find any link to this document.
But I did find an Uncyclopedia article on Stanley Random Chess. Uncyclopedia is a parody of Wikipedia, which is full of falsehoods written as humor. Checking who wrote the first version of the Wikipedia article, it is in fact Gregory Topov, the author of this page. I consider this an admission that Stanley Random Chess is a hoax.
Playing Online
This section talks about playing it on schemingmind.com, the very site the article is hosted on, but it does not include a link for actually playing it online.
Given that full documentation for the game exists only in fictional documents, the author of this page wrote an Uncyclopedia article on this game, and I cannot find anyplace to actually play it online despite claims that it can be played online, I conclude that this game is a hoax.
I will quote some excerpts from the linked document that suggest it is a hoax:
Even the page linked to does not describe the full rules of the game.
A Google search for this book did not turn up any links to it. It apparently does not exist. All that turned up were the page linked here and copies of it.
When I searched Google for "Official Stanley Random Chess Handbook", I did not find any link to this document.
But I did find an Uncyclopedia article on Stanley Random Chess. Uncyclopedia is a parody of Wikipedia, which is full of falsehoods written as humor. Checking who wrote the first version of the Wikipedia article, it is in fact Gregory Topov, the author of this page. I consider this an admission that Stanley Random Chess is a hoax.
This section talks about playing it on schemingmind.com, the very site the article is hosted on, but it does not include a link for actually playing it online.
Given that full documentation for the game exists only in fictional documents, the author of this page wrote an Uncyclopedia article on this game, and I cannot find anyplace to actually play it online despite claims that it can be played online, I conclude that this game is a hoax.