I don't think I understand how you can have a standard deviation of a mean σx̅.
x̅ is all the sum of all values divided by the sample size n. It's a single value.
How can there be a standard deviation σ for a single value? Does it mean σ of the sample it came from?
Imagine you sample the population many times. Each time you have a new sample of n observations and you compute its mean. That mean would be somewhat different for each sample. The standard deviation of this distribution is $\sigma_\bar{x}$.
$\sigma_\bar{x}$ would be zero only if n is so large that it covers the entire population or if the variance of the population itself is zero.
(Bootstrapping is one way to estimate $\sigma_\bar{x}$ but much more often it is estimated analytically by $\sigma_\bar{x}=\dfrac{\sigma_x}{\sqrt{n}}$.)
You can encounter a standard deviation when bootstrapping the mean.
For example let's say you want to have an estimation of the robustness of your mean (which is a statistic) calculated on N observations.
One way to do that is to randomly pick with replacement N individuals and then calculate the mean.
Now if you do that B times (for example B = 999), you will obtain a distribution of your mean statistic and thus be able to calculate its standard deviation.
Why would you want to do that ? To have a more robust statistic in case of new observations or outliers.
Assuming you intend sample mean then yes you can have a standard deviation. What happens if you take another sample? Will it have the exact same mean? It probably won't. Therefore, across samples you will have variability and if you have variability you have a standard deviation.
On the other hand, you don't have a standard deviation of the population mean, just of the population.
Therefore, it depends on the intent in what you're reading what the standard deviation refers to. Often the standard deviation of the sample mean is called the standard error, but they're the same thing.
