In sampling literature and causal inference literature, there usually is a distinction made about how to view observed data. The first is usually to view some observed data as having come from a finite sample, that is, the same is the entire population. The second is to view the observed data as a random sample from some hypothetical infinite-sized population.
I am wondering what deeper implications are behind thinking about a random sample from a infinitely sized population. Is the infinite part refer to the fact we have convergence results at our disposal? Why not assume a very very large, but finite result? Such that if $N$ is our sample size, then we require the population size $M$ to be $N << M < \infty$.
I feel there is a deeper intuition to assuming a random sample from an infinite population that fundamentally differs from the finite population. What might that be?