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The typical setting for least squares regression (or over-determined linear system) for $Ax=b$ is to solve $x$ given $A$ and $b$. In other words, $A$ and $b$ are fixed when we solve the problem.

My question is that is there any application that $b$ is changed by users (i.e., $b$ is interpreted as a query) while $A$ is fixed?

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In least squares regression, the problem that you solve is:

$X^T X \beta = X^T y$

where $\beta$ are the parameters, $X$ the regressors and $y$ the predicted variable. If you want, in your notation $A$ to be constant and $b$ to change, you need to have constant regressors and change the predicted variable $y$. I don't think there are applications that the regressors are not changed but the predictors do change, but if you want to find one it will probably be in online learning.

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