The reason CDFs are so useful for random variables is because a random variable's distribution (a Borel probability measure on $\mathbb{R}$) is uniquely determined by its cumulative distribution function; if you know the CDF, then you (in principle) know everything there is to know about the distribution.
That is, if $\mu$ and $\nu$ are two Borel probability measures on $\mathbb{R}$ and
$$
\mu((-\infty, x]) = \nu((-\infty, x])
$$
for every $x \in \mathbb{R}$ (equality of CDFs), then in fact
$$
\mu(B) = \nu(B)
$$
for every Borel subset $B \subseteq \mathbb{R}$ (equality of measures).
The general fact behind this result is the following corollary of Dynkin's $\pi$-$\lambda$ Theorem:
Theorem.
Let $\mu$ and $\nu$ be measures defined on a measurable space $(\mathcal{X}, \mathcal{B})$.
Let $\mathcal{E}$ be a subset of $\mathcal{B}$ such that
- $\mu(E) = \nu(E)$ for every $E \in \mathcal{E}$
- $\mathcal{E}$ is a $\pi$-system (i.e., if $E, F \in \mathcal{E}$, then $E \cap F \in \mathcal{E}$)
- $\mu$ is $\sigma$-finite on $\mathcal{E}$ (i.e., there exists a sequence $E_1, E_2, \ldots \in \mathcal{E}$ such that $\bigcup_{n=1}^\infty E_n = \mathcal{X}$ and $\mu(E_n) < \infty$ for all $n$)
Then $\mu(E) = \nu(E)$ for all $E \in \sigma(\mathcal{E})$.
The correspondence between Borel probability measures on $\mathbb{R}$ and CDFs comes from the fact that the Borel $\sigma$-algebra on $\mathbb{R}$ is generated by the set
$$
\mathcal{E} = \big\{(-\infty, x] : x \in \mathbb{R}\big\},
$$
which is a $\pi$-system on which every probability measure trivially satisfies condition (3) of the Theorem above.
It is very convenient that $\mathcal{E}$ (which is a set of subsets of $\mathbb{R}$) can be very naturally parameterized by $\mathbb{R}$ (via the map $x \mapsto (-\infty, x]$).
Following this idea, if you want to generalize CDFs to functions that determine probability distributions (AKA probability measures) on some more general measurable space $(\mathcal{X}, \mathcal{B})$, then all you need to do is find a subset $\mathcal{E}$ of $\mathcal{B}$ that is a $\pi$-system and that contains a sequence of sets whose union is $\mathcal{X}$.
For example (you already mentioned this in the question), you can consider $\mathcal{X} = \mathbb{R}^n$, $\mathcal{B} = \mathcal{B}(\mathbb{R}^n)$, and
$$
\mathcal{E} = \big\{(-\infty, \mathbf{x}] : \mathbf{x} \in \mathbb{R}^n\big\}
$$
(where $(-\infty, \mathbf{x}] = \prod_{i=1}^n (-\infty, x_i]$ for each $\mathbf{x} = (x_1,\ldots,x_n) \in \mathbb{R}^n$) to conclude that a Borel probability measure $\mu$ on $\mathbb{R}^n$ is uniquely determined by its "CDF":
$$
\mathbf{x} \mapsto \mu((-\infty, \mathbf{x}]).
$$
