Resampling small datasets - Issue of overcounting? Hypothetical situation here, as techniques like bootstrapping often fail for very small datasets.
Taking bootstrapping as an example nonetheless. we can easily compute the number of possible bootstrap (re)samples.
A nice answer is given by @whuber here:
Amount of Possible Bootstrap Samples
Suppose for a moment that bootstrapping is perfectly valid for low sample sizes. Now, say we have $n = 5$. From the posted solution above, it is found that there are a total of 126 possible boostrap resamples that can be drawn.
In bootstrapping, we usually take a large number of replications (10000 for instance). Taking this many replicates for such a small dataset, like the one above, seems bizarre, since resamples will be counted multiple times.
Question: Does this really matter? If yes. what is the effect on inference?

The issue with small sample sizes is not that you will repeat bootstrap samples but that the original small sample might not be so representative of the population.
Let’s obtain a sample of coin flips from a fair coin, so the true population is $Binom(1, 0.5)$, and let’s use your small sample size of $n=5$. In R…
set.seed(314) # For pi
x <- rbinom(5, 1, 0.5)

I get four $0$s (heads) and one $1$ (tails), which means a $20\%$ chance of tails, rather than the correct $50\%$. When we go to bootstrap this sample, we are telling the bootstrap procedure to sample from a $Binom(1, 0.2)$ distribution, which is quite a bit different from the true $Binom(1, 0.5)$ population.
When the sample size is larger, we are less likely to have a sample that is so drastically different from the population.
