Yes, this is very much a real chess problem. Interestingly, it seems that the name for it is 'The Obelisk."
I have found mention of it in a Google scanned book. The piece of literature is called "American Chess Review, Volume 1, Issues 1-6" and it can be read in it's entirety for free here on Google as an eBook.
The book is from 1886, just four years after the mentioned publication in Bretano's Chess Monthly. On page 99 it reads, as quoted, : "and the “Obelisk' (mate in 1,220 moves, compelling three successive knight's tours!) contributed by the genius of Mr. J. N. Babson to the late Brentano's Chess Monthly." It is listed with a few other mysterious longmovers that I am yet to do any research on.
The set up of the postion, which is descriptive notation (Wikipedia article) matches up with what you have shown. Here is the quote: "THE OBELISK: White—K at Ki Q at Q; R at Q B, KB: B at Q 6, K 6: S at Q 7, K 7; Pat Q 2, 3, 4, 6, K2, 3, 4, 6. Black—K at K, S at Q. White to play and mate in 1220 moves, after compelling black to make three complete and successive tours."
Here is a nice little picture of it all.
Note that the 'S' stands for Knight: this is the German notation, and German culture has had an impact on American culture. It lines up exactly with your diagram, except that the bishops are listed one square as the pawns are listed. I take this as a printing mistake.
[Title "The Obelisk, Mr. J. N. Babson, 1882"]
[FEN "3nk3/3NN3/3PP3/3BB3/3PP3/3PP3/3PP3/2RQKR2 w KQ - 0 1"]
So it is very much a real chess problem. A couple of other places I find it mentioned in (albeit in book previews) are on page 205 of Wonders and Curiosities of Chess, Irving Chernev, 1974: Wonders and Curiosities (the link is to Google Books.) There is also the claim of it appearing in "The Complete Chess Addict" and ""The Even More Complete Chess Addict" on this ChessChat Forums page. I'll go indepth on that soon. Rosie F, in a helpful comment, says that as well: "Mike Fox & Richard James copied it in their The Complete Chess Addict (pub.Faber 1987), p.174, but don't give any clue as to a solution."
However, no where at all, it there a mention of a solution. Unless somebody can get their hands on a original/reprint, several of which can be found for sale around the Internet with a quick search, their is officially no known one.
Something I believe that may confirm is something very small that I noticed. Here is a link to the problem on Yet Another Chess Problem Database. It is called yacpdb for short. There, two sources for the problem are listed (which I will track down soon if I can), one of which is the Problemiste PBM Collections.
The interesting fact is that if actually a collection made by The Problemist, a famous chess column or whatever it is called. The yacpdb references section lists these words from the Problemist: "Remark text: Note de Le Lionnais : "Nous n'avons pas pu découvrir la solution de ce problème ni même nous assurer qu'elle n'a pas été démolie."
Translated, it means this: "Remark text: Note from Le Lionnais: "We have not been able to discover the solution of this problem nor even convince ourselves that it is not cooked.
So, while the problem is real, there is no known solution.
As for your other two questions, it does seem to be somewhat widely known. It is on this chess.com page in the comments on a forum post there.
I have no answer for your second question, unfortunately.
I think that there is a small possibility that there actually IS NO SOLUTION, and that this is Babson's greatest ever JOKE PROBLEM.