Consider the two following versions of a random dynamical system:
$\mathbf{1.}:$ Let $(\Omega, \mathcal{F}, \mathbb{P}, \sigma)$ be a probability preserving dynamical system. Let $T_{\omega}: X \rightarrow X , \omega \in \Omega$ be a family of maps on some compcat metric space $(X, d).$ The random dynamical system is then created by looking at the composition $$T^n_{\omega} (x) = T_{\sigma ^{n-1} \omega} \ \circ ... \circ \ T_{\omega} (x)$$ for a point $x \in X,$ with the corresponding skew product $$T(\omega, x) = (\sigma \omega, T_{\omega} (x)).$$
$\mathbf{2.}:$ Let $(\Omega, \mathcal{F}, \mathbb{P}, \sigma)$ be a probability preserving dynamical system. For each $x \in X$, let $X_{\omega} \subset X$ be a subset of a compact metric space $(X, d).$ Let $T_{\omega} : X_{\omega} \rightarrow X_{\sigma \omega}$ be a transformation. The random dynamical system is then created on the bundle $$\mathcal{E} = \bigsqcup_{\omega \in \Omega} \omega \times X_{\omega}$$ with the skew product $S: \mathcal{E} \rightarrow \mathcal{E}$ and $$S(\omega, x) = (\sigma \omega, T_{\omega}(x)).$$ Similarly, $$T^n_{\omega} (x) = T_{\sigma ^{n-1} \omega} \ \circ ... \circ \ T_{\omega} (x).$$
Are these random dynamical systems doing the same thing? In other words, are these equivalent? I am not totally sure what the second system is doing.
Clearly the constructions are the same if $X_\omega=X$ for all $\omega$, right?
It may be helpful to note that this is a very common construction that all of us should be used to. Namely, let $f\colon M\to M$ be a diffeomorphism of a smooth manifold $M$. Then each derivative $d_xf\colon T_xM \to T_{f(x)}M$ takes a tangent space $T_xM$ to another tangent space $T_{f(x)}M$. This is somewhat awkward to work with since the tangent spaces vary.
The "solution" can be to define a map $F\colon TM\to TM$ (usually called extension, cocycle, etc) by $$ F(x,v)=(f(x),d_xfv) $$ for $(x,v)\in M\times T_x M$. Notice the similarity to your map $S\colon \mathcal{E} \to\mathcal{E}$ defined by $$S (\omega, x) = (\sigma (\omega), T_{\omega}(x)) $$ for $(\omega,x)\in \Omega\times X_\omega$, where $X_\omega$ corresponds to $T_xM$.
In many situations we are able to identify (sometimes unavoidably using charts) all tangent spaces $T_xM$ with some fixed $\mathbb R^k$ where $k$ is the dimension of the manifold. The same may occur in your setting, depending on what is $X_\omega$ after all, which one may be able to identity, or not, with a single set $X$ in some precise manner.
This gives the answer to your question: sometimes they are equivalent, sometimes they are not.
I should also add that both constructions are very particular cases of a general construction obtained from a group action, in your case a measurable group action (which means that the map on the first component preserves the measure). Note that measurable vector bundles need not have all fibers of the same dimension.