No, not in general. For instance $GL_4(2)$ is isomorphic to the alternating group $A_8$, and the direct product of the two Klein four groups acting regularly on $1,2,3,4$ and $5,6,7,8$, respectively, is abelian of order $16\gt 2^4-1$.

I would expect your result to be true if $M$ is a $p'$-group. At ant rate, Maschke + Schur easily show that $\lvert M\rvert\le p^n-1$ when $M$ is an abelian $p'$-group.

No, not in general. For instance $GL_4(2)$ is isomorphic to the alternating group $A_8$, and the direct product of the two Klein four groups acting regularly on $1,2,3,4$ and $5,6,7,8$, respectively, is abelian of order $16\gt 2^4-1$.

Actually, for any prime $p$, there is an abelian subgroup of order $p^4$ in $GL_4(p)$, just take the matrices of the form $\begin{pmatrix} E & A\\ 0 & E\end{pmatrix}$, where $E$ is the $2\times 2$ identity matrix, and $A$ an arbitrary $2\times 2$ matrix.

Generalizing to bigger $n$, there are more drastic counterexamples.

I would expect your result to be true if $M$ is a $p'$-group. At ant rate, Maschke + Schur easily show that $\lvert M\rvert\le p^n-1$ when $M$ is an abelian $p'$-group.