If by "$=$" you mean $\beta$-equality, then the answer is yes, $MX=X$ for all $X$ is a stronger property than $MM=M$.
For example, let
$$A := \lambda a.aa(aa)$$
(to save parentheses, I am using the standard left-associative notation for application; in your notation, the above term would be $\lambda a.a(a)(a(a))$) and take
$$M := AA.$$
We clearly ...