Translating to the language of econometrics, I think the point of this paper is that oftentimes, the instrument that you're using has errors in its measurement,

Translating to the language of econometrics, I think the point of this paper is that oftentimes, the instrument that you're using has errors in its measurement, or is just a proxy for the 'real' instrument that you'd like to use. If that's the case, when IV is interpreted as a two-stage regression, you might still have covariates that are correlated with the residuals in the second stage.