If the standard error is the standard deviation of the distribution of the sample means, what would be the point/meaning behind calculating the standard error of a single sample?

The standard error tells you how accurate the mean of any given sample is likely to be compared to the true population mean. When the standard error increases, i.e. the means are more spread out, it becomes more likely that any given mean is an inaccurate representation of the true population mean. In simple words, The lesser the standard error, the more accurate our predictions about the population mean and vice versa.