The point is that, if you set $x_i = {u_i}^2$ where $u_i>0$, this becomes a diffeomorphism of $M$ with itself with the property that, in the $u$-coordinates, the Shahshahani metric becomes $$ g = 4({u_1}^2+\cdots+{u_n}^2)\bigl({\mathrm{d}u_1}^2+\cdots + {\mathrm{d}u_n}^2\bigr). $$ Clearly, this metric is just the flat metric in the $u$-coordinates on the positive orthant times the squared '$u$-distance' from the origin. When $n\ge2$, any rotation in the $u$-coordinates would preserve the metric on the entire $\mathbb{R}^n$, but it wouldn't preserve the positive orthant, which is $M$. So, globally on $M$, you are looking at the rotations (in the $u$-coordinates) that preserve the positive orthant, and this is just the permutations in the $u_i$, which is the permutations in the $x_i$.

Note: When $n=2$, this is actually a flat metric: If one sets $z = u_1 + i\,u_2$, then we have $g = 4 z\bar z\, \mathrm{d}z\circ\mathrm{d}\bar z = \mathrm{d}(z^2)\circ\mathrm{d}\bar z^2 = \mathrm{d}w\circ\mathrm{d}\bar w$ where $w = z^2 = (u_1 + i\,u_2)^2$. Thus, the Lie algebra of Killing fields has dimension $3$ instead of $1$. However, when $n>2$, the metric is not flat, and the Killing fields are the generators of the obvious $\mathrm{SO}(n)$-action, as can be seen by direct calculation or conversion to 'polar coordinates'.

The point is that, if you set $x_i = {u_i}^2$ where $u_i>0$, this becomes a diffeomorphism of $M$ with itself with the property that, in the $u$-coordinates, the Shahshahani metric becomes $$ g = 4({u_1}^2+\cdots+{u_n}^2)\bigl({\mathrm{d}u_1}^2+\cdots + {\mathrm{d}u_n}^2\bigr). $$ Clearly, this metric is just the flat metric in the $u$-coordinates on the positive orthant times the squared '$u$-distance' from the origin. When $n\ge2$, any rotation in the $u$-coordinates would preserve the metric on the entire $\mathbb{R}^n$, but it wouldn't preserve the positive orthant, which is $M$.

When $n=2$, this is a flat metric: Set $z = u_1 {+} i\,u_2$. then $g = 4 z\bar z\, \mathrm{d}z\circ\mathrm{d}\bar z =\mathrm{d}w\circ\mathrm{d}\bar w$ where $w = z^2 = (u_1 + i\,u_2)^2 = ({u_1}^2{-}{u_2}^2)+ i\,(2u_1u_2)$. Thus, $w:M\to\mathbb{C}$ isometrically embeds $(M,g)$ as the upper half-plane in $\mathbb{C}$ when $\mathbb{C}$ is given its standard metric. The global isometries are the translations by a real number in the $w$-coordinate together with reflection in the imaginary axis. Meanwhile, the Lie algebra of Killing fields has dimension $3$ instead of $1$.

However, when $n>2$, the metric is not flat, and the Killing fields are the infinitesimal generators of the obvious $\mathrm{SO}(n)$-action, as can be seen by direct calculation or conversion to 'polar coordinates'. Meanwhile, globally on $M$, you only have the rotations and reflections (in the $u$-coordinates) that preserve the positive orthant, and this is just the permutations in the $u_i$, which is the permutations in the $x_i$.

The point is that, if you set $x_i = {u_i}^2$ where $u_i>0$, this becomes a diffeomorphism of $M$ with itself with the property that, in the $u$-coordinates, the Shahshahani metric becomes $$ g = 4({u_1}^2+\cdots+{u_n}^2)\bigl({\mathrm{d}u_1}^2+\cdots + {\mathrm{d}u_n}^2\bigr). $$ Clearly, this metric is just the flat metric in the $u$-coordinates on the positive orthant times the squared '$u$-distance' from the origin. When $n\ge2$, any rotation in the $u$-coordinates would preserve the metric on the entire $\mathbb{R}^n$, but it wouldn't preserve the positive orthant, which is $M$. So, globally on $M$, you are looking at the rotations (in the $u$-coordinates) that preserve the positive orthant, and this is just the permutations in the $u_i$, which is the permutations in the $x_i$.

The point is that, if you set $x_i = {u_i}^2$ where $u_i>0$, this becomes a diffeomorphism of $M$ with itself with the property that, in the $u$-coordinates, the Shahshahani metric becomes $$ g = 4({u_1}^2+\cdots+{u_n}^2)\bigl({\mathrm{d}u_1}^2+\cdots + {\mathrm{d}u_n}^2\bigr). $$ Clearly, this metric is just the flat metric in the $u$-coordinates on the positive orthant times the squared '$u$-distance' from the origin. When $n\ge2$, any rotation in the $u$-coordinates would preserve the metric on the entire $\mathbb{R}^n$, but it wouldn't preserve the positive orthant, which is $M$. So, globally on $M$, you are looking at the rotations (in the $u$-coordinates) that preserve the positive orthant, and this is just the permutations in the $u_i$, which is the permutations in the $x_i$.

Note: When $n=2$, this is actually a flat metric: If one sets $z = u_1 + i\,u_2$, then we have $g = 4 z\bar z\, \mathrm{d}z\circ\mathrm{d}\bar z = \mathrm{d}(z^2)\circ\mathrm{d}\bar z^2 = \mathrm{d}w\circ\mathrm{d}\bar w$ where $w = z^2 = (u_1 + i\,u_2)^2$. Thus, the Lie algebra of Killing fields has dimension $3$ instead of $1$. However, when $n>2$, the metric is not flat, and the Killing fields are the generators of the obvious $\mathrm{SO}(n)$-action, as can be seen by direct calculation or conversion to 'polar coordinates'.