System Identification using LMS Adaptive Filter
Because an LMS estimator will, over time, "average out" uncorrelated zero-mean noise. It's pretty much in the name.

But yes, you're right, there is a noise component in the estimate; one of the qualities of an estimator is how little the noise variance influences the estimate variance after a given length of observation.

That's the case, however, for all estimators: you measure signal + noise, and you estimate parameters from that. The parameters must be somewhat noisy; otherwise, there's something broken with your noise model.