Two expressions with similar arrangement of coefficients, does it imply that the variables are same? If $a|x|^2 + b|x| + c = a|y|^2 + b|y| +c$, then is it always $|x|=|y|$ ?

Or similarly,

if  $a|x|^n + b|x|^{n-1} + c|x|^{n-2} = a|y|^n + b|y|^{n-1} + c|y|^{n-2}$, then is it $|x| = |y|$ ?

More generally, does it happen that, 

when,
$a|x| + b|y| + c|z|  =  a|m| + a|n| + c|p|$ 

[where, a,b,c are constants. Others are variables]

then,  $|x|=|m|$ , $|y| = |n|$ and $|z| = |p|$ ?
In other words, you're asking whether the functions
$$ x \mapsto ax^2+bx+c \\
(x,y,z) \mapsto ax+by+cz $$
are injective. No, they are not, in general.

$ax^2+bx+c$ can be injective, namely if (but only if) $a=0$ and $b\ne 0$.

$ax+by+cz$ is never injective no matter what $a$, $b$, and $c$ are.
In the first case, the answer is clear: yes. But not only. You have in general two solutions. Let be:
$$s=x-y$$
$$t=x+y$$
$$a(x^2-y^2)+b(x-y)=0$$
$$a(x-y)(x+y)+b(x-y)=0$$

Then $ast+bs=0$, $s(at+b)=0$ That bring us two solutions: $s=0$, $x=y$ and $t=\frac{-b}{a}$, $y=\frac{-b}{a}-x$