Can a variable affected by a random variable be also considered a random variable by extension? I'm currently learning about instrumental variables analysis.  One of its assumptions – the exclusion restriction – states that the instrument affects the outcome only through the exposure variable, and that the instrument is a random occurrence in nature.  It basically emulates a randomized controlled trial wherein the instrument determines the exposure.
If it were an actual RCT, then treatment assignment is random.  But since in instrumental variables analysis, the instrument is random and exogenous (supposedly) and that it determines the exposure, would exposure in such context be considered 'random' as well?

Yes. A variable that is a measurable function of a random variable can itself again be considered a random variable.
The definition of a random variable is that it is a measurable function from a probability space $(\Omega, \mathcal F, p)$ to $\mathbb R$. So if $f:\Omega\to\mathbb R$ is a random variable and $g:\mathbb R\to\mathbb R$ is the variable we are considering, then the composition $g\circ f$ is again a measurable function $\Omega\to\mathbb R$. So $g$ is considered a random variable in the sense that its composition $g\circ f$ with $f$ is a random variable.
Thus, if in a model, we separate all the variables into exogenous and endogenous ones, all variables are considered random variables, including the endogenous ones, even though they are "only" random variables by composition with other random variables.
Hence, in your case, the exposure variable is also considered a random variable.
