Suppose I define the conditional expectation of a random variable $X$ with respect to the $\sigma$-algebra $\mathcal{M}$ denoted $E[X|\mathcal{M}]$ on some probability space $(\Omega, \mathcal{F}, P)$.
Is there a natural associated probability measure on $\mathcal{M}$ that goes along with $E[X|\mathcal{M}]$? This would be probability measure which in some sense contains the information only contained in $X$.
It isn't really measures which are thought of as containing information, but rather $\sigma$-algebras. Indeed, probability measures are a somewhat disjoint concept from the concept of a $\sigma$-algebra.
For any random variable, the $\sigma$-algebra $\sigma(X)$ generated by the events $[X\in E]$ for $E\subset\mathbb R$ open is a natural $\sigma$-algebra associated to the r.v. $X$ in the sense that $\sigma(X)$ is the smallest $\sigma$-algebra $\mathcal F$ such that $X$ is $\mathcal F$-measurable. Then, if you have any probability measure $P$ on $\Omega$, you are free to consider the restricted probability space $(\Omega,\sigma(X),P|_{\sigma(X)})$. The discussion of this paragraph applies to r.v.'s of the form $E[X\mid\mathcal F]$.
$\sigma$-algebras are indeed very detailed objects, so it can be hard to see the forest for the trees when it comes to the "containing information" perspective. The most basic way to see it is to consider a r.v. $X$ that takes only finitely many values $a_1,\dots,a_n$. Then all the information we could want to know about $X$ is contained in the events $[X=a_1],\dots,[X=a_n]$ since knowing which of these level sets our sample $\omega\in\Omega$ is in tells us everything about $X$ from the perspective of probability. There are more complicated events we can fashion from these $n$ events like $[X\in\{a_1,a_2,a_5\}]$, but all of these events are captured by the idea of forming the $\sigma$-algebra generated by the events $[X=a_1],\dots,[X=a_n]$.
In a more complicated situation, we may be considering several r.v.s $X_1,\dots,X_n$, or even a continuum of r.v.s like $(X_t)_{t\in[a,b]}$ (as in the study of continuous-time random processes). The basic elements to keep in mind are always the level sets $[X < r]$, $r\in\mathbb R$, as these are essentially the simplest events telling us fundamental information about the variables $X$. Then when we want to ensure we are capturing all the detailed events or "information" about $X$, we pass to the $\sigma$-algebra generated by the level-set events $[X<r]$, $r\in\mathbb R$.