There's truth, however it is not just "this rule" but many similar rules that have existed before the born of modern Chess.
I know the knight jump of the King (at its first move) is known as "Indian Castling" because coming form India and it is still in use in nowadays Cambodian Chess (Ouk Chatrang) together with a similar Firzan jumping rule. In medieval Europe there were three major variants, from what I know, one that we can call "English-French Chess" or just Medieval Chess, then German Chess and finally Lombard Chess (from Italy). Any of them had some special "jumping move" for both Firzan and King, or even the two pieces together in some version/variant, I can't remember exactly.
My information is not exhaustive, nor surely accurate. Consider reading "A History of Chess" written by Murray in 1913 (and still the most complete guide on the subject), I think it's around 500 pages I guess. It covers all traditional variants of Chess all over the world and there's A LOT on Shatranj/Chaturanga and medieval European Chess as well.
I've been playing Chaturanga on here for a while, and I figured I would see if there are any videos of this game on youtube. I found this video, which is listed amongst many other ancient and regional variations of chess on the same person's channel:
https://www.youtube.com/watch?v=LIb1yx6CZqQ (4:40 mark)
At the selected time spot, he claims "they did not have castling like we do today, some regional variants would have another form of castling, such as the king making a knight's move early in the game" (presumably the king's first move)
I have seen this claim repeated in some of his other regional/historic variants of chess. While he may be an amateur, he does appear to have studied the game, owning many antique chess boards, visiting many chess museums, compiling different resources for each variant and publishing booklets for them. I have to assume he didn't just make this claim up, that it had to have come from somewhere. It seems logical that before castling was invented, there was some type of move already floating around where the king could move more than one space, and the king moving like a knight would transpose into a castle like position after the rook is moved.
So, my question is: is anybody familiar with this rule? Does anybody know where this rule comes from? Is this a misconception or is there truth to it?