Le Maohua, “On Mersenne Numbers” [in Chinese], Journal of Jishou University (Natural Science Edition) 20(1) (March 1999): 17–19, shows that the squarefree part of $2^p - 1$, with $p \ge 11$ prime, is greater than $(\pi{}p/log\thinspace{}p)^2$.
(From here to end added in 2023:) Since Le’s paper is not widely available, I will attempt some explanation of it, working from notes in English that were provided to me privately and which I am not in a position to share.
Le was apparently the first to notice the relevance to this problem of a paper of Wilhelm Ljunggren, “Über die Gleichungen $1 + Dx^2 = 2y^n$ und $1 + Dx^2 = 4y^n$,” Det Kongelige Norske Videnskabers Selskab Forhandlinger 15(30) (1942): 115–118, on the Diophantine equation $1 + Dx^2 = 4y^n$. The case $D = a$, $x = b$, $y = 2$, and $n - 2 = p$ (with $p$ prime) corresponds to a Mersenne number factorization in the form proposed in the original post above, $M_p := 2^p - 1 = a \cdot b^2$. Now if $h(\cdot)$ represents the class number of the imaginary quadratic number field $\mathbb{Q}\left(\sqrt{\cdot}\right)$, then by Gauss’s theory of classes, $h_p(-M_p) = h(-b^2a) = h(-a)$. Ljunggren showed that $p - 2$ divides the class number $h(-a)$, which is in itself a notable result.
After verify the nonexistence of a square divisor in published factorizations of $M_p$ for the small cases $11 \le p < 101$, Le uses Ljunggren’s result and estimates of the class number by Louboutin and others to establish a lower bound on the squarefree part of $M_p$ in the remaining cases.