Can you give examples? It seems to me that in my cases, I never use X='temperature' as a predictor, but rather X='temperature reported by thermometer', and this is of course always known with 100% accuracy. In a predictive context, that's all we need. Are you referring to a social science context?

Here's an easy one to conceptualize: A good friend of mine is a state hydrologist. He crawls over streams with a ladder-like platform and at increments lowers a depth gauge to eventually estimate the stream's volume flow rate. For a number of reasons (including wild weather conditions) the cross stream increment is known only to an inch or so, that's the uncertainty in X, the depth gauge reading has another uncertainty in Y. These are applied to a function where the slope, via Deming regression, gives him an estimate of the flow rate. By the bye, most lab thermometers i've dealt with have an accuracy of about 0.1°C

Looks closer to support vector machine and its regression capabilities.

Although, linear regression, logistic regression, svm regression, are all almost the same thing, the loss function is different, all can be trained with gradient descent.