I am getting $L_1$ regular and $L_2$ CFL.
$\Rightarrow M=\left \{ xyx |x,y\epsilon \left ( a,b \right )^*\right \}=(a+b)^*$
$L_1=L-M=L\cap M^{'}=L \cap \phi=\phi$ which is regular
$L_2$is CFL as CFL are closed under concatenation and nothing is given about $L$