Using the information provided by Flounderer, I think I finally understand.
MannKendall into R, I indeed get that the function is really calling the Kendall function as Kendall(1:length(x), x).
The Kendall function requires various considerations. In my case, the values of X have no ties.
Case 1: No Ties:
Subcase 1: length(x)<9: In this case the probabilities are computed using a recurrence relation described in kaarsemaker and van wijngaarden 1953.
Subcase 2: length(x)>=9: In this case the probabilities are calculated using a modified edgeworth series. What is important to note here is that I don't think any asymptotics are being called, but rather you are approximating the cumulative probability density by truncating an infinite series. This method is presented in Best and Gipps (1974). In the paper, they mention that the maximum error for any probability is 0.0004.
Case2 : There are ties:
This is the case in which asymptotics are called in order to use a normal approximation. The details are explained in the R CRAN package documentation.