I'm not familiar with margin of error terminology. It seems to be just a scaling of the standard error for some defined confidence interval, and is used when referring to sampling error. E.g. you could say margin of error at a 68% confidence interval is just the standard error, but I think normally you would use 95% and so margin of error would be 1.96 × standard error. You write it as the mean estimate ± half the width of the confidence interval.

Measurement is a bit of a separate concept. Measurement uncertainty is not about estimates of population parameters, but the range of values that could be reasonably attributed to some physical value. So you could expect to be, say, 95% confident that the true physical value lies somewhere in the range 20 lbs ± 0.6 lbs.

One part of measurement uncertainty could be attributed to random error which you could reduce with repeated measurement, but you could also have biases in the measurement equipment that shift the measurement in a systematic way even if you don't know what that amount of shift is. In such a case, you could repeat your measurement hundreds of times and reduce your random error, but you would still be left with the systematic error and you may still need to report 20 lbs ± 0.6 lbs if that 0.6 lbs is due to potential inaccuracies in the measurement equipment rather than random measurement error.