What you've done so far does not suffice to show that $T$ is not linear.
To show that $T$ is not linear, you have to show that one of the following two properties does not hold:
$\ \ \ \ $1) $T(a{\bf x})= aT({\bf x})$ for all scalars $a$ and all vectors $\bf x$.
$\ \ \ \ $2) $T({\bf x}+ {\bf y})=T({\bf x})+T({\bf y})$ for all vectors $\bf x$ and $\bf y$.
You've noticed $\tag{ 3 }T(3,2,6)=T(6,2,3)=(2,3,6).$
Now notice, by the definition of $T$, that $T\bigl( (3,2,6)+(6,2,3)\bigr)=T(9,4,9)=(4,9,9).$ But from $(3)$ $ T(3,2,6)+T(6,2,3)=(2,3,6)+(2,3,6)=(4,6,12)\ne(4,9,9). $
So, 2) fails and, thus, $T$ is not linear.
Note that you need only verify that one of the two properties fails. Here, you could have shown that property 1) fails instead: $T(1,2,3) =(1,2,3)$ but $T(-1,-2,-3)=(-3,-2,-1)\ne(-1)T(1,2,3).$