If $f(x)=|x|$ for all real $x$, then the function $f$ is of degree $2$. However, if $f=P\circ g$ for some polynomial $P$ of degree $2$, then 
$$g(x)=\frac{-b\pm\sqrt{b^2-ac+a|x|}}a
$$
for some real $a\ne0,b,c$ and all real $x$, and such a function $g$ cannot be everywhere increasing, because the function $\mathbb R\ni x\mapsto b^2-ac+a|x|$ is not monotonic.