Why the standard error of the mean never gets to zero even when the sample has the same size as the population? Why, when we calculate the standard error of the mean for a hypothetical sample with the same size as the population, the value is not equal to zero, if the sample in this case is identical to the population?

If the sample has the same size as the population, either some units from the population are sampled multiple times, or the sample is in fact the whole population.
In the former case, the sample mean provides an estimate of the population mean, in which case a different sample would likely yield a different estimate of the population mean. Thus, the standard error of the estimate should be greater than zero (unless the sample has zero variance, of course).
In the latter case, there is no need to estimate the population mean, because the population mean can simply be calculated. This statistic has a variance of zero; it will not vary if we sample again using the same sampling procedure. Thus, the standard error is zero in that case.
