Is a Random/Stochastic process always dependent on 'time', or can it be dependent on any other 'unit'?
For instance, suppose,
We roll a dice whenever a soccer match takes place in a major league in a country in Europe.
Now, a soccer match can take place at any time in Europe dependent on the schedule of the league of in a country.
Yes. Here are a few thoughts of mine. I am not sure whether it really qualifies as an answer.
A stochastic process $X=(X_i)_{i\in I}$ for some ordered set $I$ is nothing more than a collection of random variables. Examples of index sets $I$ include $[0,\infty)$, which is often interpreted as continuous time, or $\mathbb{N}$ or $\mathbb{N}_0$, which might be interpreted as discrete time.
Examples not necessarily involving time could be random fields. For an image given by a unit square $[0,1]^2$, $X_{(y,z)}$ could be the (random) salt-and-pepper noise in the point $(y,z)\in[0,1]^2$. See e.g. https://arxiv.org/pdf/1609.06341.pdf.
Your question is connected to the fact that we often may view stochastic processes in two distinct ways: as random functions $X : \omega \mapsto X(\omega)$ or as collections of random variables $(X_i)_{i\in I}$. Filtrations are e.g. related to the second view rather than the first. Personally, if a problem only requires me to think of $X$ as a random function, as in the example given above, and not as a collection of random variables, I would avoid calling $X$ a stochastic process, because I am not studying it as a process $X : (\omega, i) \mapsto X_i(\omega)$.