The existence of such a biholomorphism is rather rare. For example, as a simple exercise you can check that already for a conic in $\mathbb CP^2$ such a biholomorphism does not exist. At the same time, in the case $M$ is "exceptional" in $X$, for example, $X$ is obtained by a blow up from $X'$ at a point $x'\in X'$ and $M$ is the exceptional divisor, the situation is the often the one you want (I don't specify purposefully here what "exceptional" means).
Solution to the exercise. Everything follows from the fact that each holomorphic map from $\mathbb CP^1$ to itself has at least one fixed point. Indeed if a neighbourhood of a conic in $\mathbb CP^2$ were biholomorphic to a neighbourhood of $O(4)$ bundle over $\mathbb CP^1$ then there would be a holomorphic family of lines in $\mathbb CP^2$ parametrised by the points of the conic passing through the corresponding point of the conic and transversal to it. Such a family of lines would induce a holomorphic self-map of the conic without fixed point (take the second intersection of each line with the conic). Contradiction.
PS. I remember a theorem that states that the only smooth hypersurface in $\mathbb CP^n$ that has a neighbourhood isomorphic to a neighbourhood of a zero section in a line bundle is a linear $\mathbb CP^{n-1}$ (but I forgot the reference).