user298: Wikipedia quotes:

The Indian Defences by g6 coupled with d6, or b6 coupled with e6, were
  largely taught to European players by the example of Moheschunder and
  other Indians, to whom the fianchetto developments were a natural
  legacy from their own game.

(The "their own game" in this instance meaning the older variant.)
However, my understanding of Shatranj/Chaturanga rules do not allow fianchetto movements at all.
Is there another older variant that allows fianchetto movements that these people played?

user167: The page you linked to mentions that "Mohishunder originally played traditional Indian chess, in which pawns did not have the option of moving two squares from the starting row." I think that reference to traditional Indian chess refers not all the way back to shatranj/chaturanga, but rather to this 18th and 19th century variant, which has some significant differences from chess as we know it today, but seemingly does not in any way preclude a fianchetto. And since those rules do preclude any advance of a pawn by two squares, it seems reasonable that systems combining g6/d6 or b6/e6 would naturally arise as methods to develop the bishops in that variant.
Updated References: As the OP noted in a comment, the wikipedia entry I linked to doesn't indicate any source material. I found another page which describes this variant under the name Hindustani Chess, and it includes a helpful pointer to the 1913 book A History of Chess, a 900-page tome written by H.J.R. Murray, which details this particular chess variant in its Chapter IV. As it turns out, it appears that Skyhorse Publishing will be reissuing the book later this year. From their blurb:

An epic work that took over a decade to complete, A History of Chess, originally published in 1913, was a historic undertaking that shattered preconceptions about the game upon its release. Over a century later, Murray’s research, in which he argues that chess originated in India, is still widely accepted by most chess historians.


298: <p>Wikipedia quotes:</p>

<blockquote>
  <p>The Indian Defences by g6 coupled with d6, or b6 coupled with e6, were
  largely taught to European players by the example of Moheschunder and
  other Indians, to whom the fianchetto developments were a natural
  legacy from their own game.</p>
</blockquote>

<p>(The "their own game" in this instance meaning the older <a href="http://www.chessgames.com/perl/chessplayer?pid=85637" rel="noreferrer">variant</a>.)
However, my understanding of Shatranj/Chaturanga rules do not allow fianchetto movements at all.</p>

<p>Is there another older variant that allows fianchetto movements that these people played?</p>
 167: <p>The page you linked to mentions that "Mohishunder originally played traditional Indian chess, in which pawns did not have the option of moving two squares from the starting row." I think that reference to traditional Indian chess refers not all the way back to shatranj/chaturanga, but rather to <a href="http://en.wikipedia.org/wiki/Indian_chess" rel="noreferrer">this 18th and 19th century variant</a>, which has some significant differences from chess as we know it today, but seemingly does not in any way preclude a fianchetto. And since those rules do preclude any advance of a pawn by two squares, it seems reasonable that systems combining g6/d6 or b6/e6 would naturally arise as methods to develop the bishops in that variant.</p>

<p><strong>Updated References:</strong> As the OP noted in a comment, the wikipedia entry I linked to doesn't indicate any source material. I found another page which describes this variant under the name <a href="http://www.chessvariants.org/historic.dir/hindustani.html" rel="noreferrer">Hindustani Chess</a>, and it includes a helpful pointer to the 1913 book <em>A History of Chess</em>, a 900-page tome written by H.J.R. Murray, which details this particular chess variant in its Chapter IV. As it turns out, it appears that Skyhorse Publishing will be <a href="http://www.skyhorsepublishing.com/book/?GCOI=60239108641190&amp;" rel="noreferrer">reissuing</a> the book later this year. From their blurb:</p>

<blockquote>
  <p>An epic work that took over a decade to complete, A History of Chess, originally published in 1913, was a historic undertaking that shattered preconceptions about the game upon its release. Over a century later, Murray’s research, in which he argues that chess originated in India, is still widely accepted by most chess historians.</p>
</blockquote>
