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?

When sampling from finite population, the standard error of the mean $\frac{\sigma}{\sqrt{n}}$ must me multiplied with the finite population correction (FPC) $\sqrt{\frac{N - n}{N - 1}}$ where $N$ is the population size and $n$ the sample size. Hence, the corrected standard error of the mean is:
$$
\frac{\sigma}{\sqrt{n}}\sqrt{\frac{N - n}{N - 1}}
$$
and because $\lim_{n\rightarrow N}\sqrt{\frac{N - n}{N - 1}} = 0$, the standard error of the mean goes to zero, as the other posts explained.
Note that if you work with the sample variance using Bessel's correction ($n-1$) then the FPC is $\sqrt{\frac{N-n}{N}}$ (see here for a derivation).
