Thank you for sending that Skyline document.
The lengths of the whiskers which are drawn on the Fold Change Bar Graph is controlled by the "Confidence level" number that is specified in the Group Comparison definition.
For your group comparison, that confidence level is set to "95%", which means that the length of those whiskers will be 1.96 times the standard deviation, because, for a normal distribution, the 95% confidence interval is always 1.96 standard deviations.
If you were to change the Confidence level in the Group Comparison settings to 68%, then you would see that the size of the whiskers in the Fold Change Bar Graph were the same as the standard error value that you see in the Grid.
It would probably be better if the column that you got when you copied data from the Fold Change Bar Graph was named "95% confidence interval" instead of "StdErr". The reason that Skyline is doing what it is doing is that the "Copy Data" does not actually know anything about group comparisons. Whenever the Copy Data sees a box and whisker plot, it always calls the whisker length "StdErr", even if that's not what the whiskers represent in this case.
By the way, there is more information about how Skyline calculates fold changes using a linear regression here: