Mr. Shah, you are calling the outputs "standard deviations" when they really are not.
You will need to get under the hood to see how the "standard deviations" are computed in both examples. I think you will find that the two functions in your examples are actually measuring different things. Happily, standard deviations feed into both metrics and so the SDs (as well as standard errors) can be recovered and compared as you are trying to do.
Start with your second example (using forecast::auto.arima(x)). This function produces prediction intervals, not confidence intervals. However, you (improperly) computed your standard deviations as though those PIs were confidence intervals. So, you will want to use a different method to recover the standard deviation (or standard error, if that is what you are really after). I think converting the PI to the related CI will do what you want and allow a direct comparison with your first example results.
If that still does not work, you will want to check to make sure that you are comparing a standard error in example 2 to the standard error that was computed in example 1, and not comparing a standard deviation to a standard error.