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In a standard Tobit, what's the relation between the standard error of the regression and the scale factor?

That's an interesting question. There are probably a lot of ways to look at this, but here's my take on it...

The scale factor is, for the normal error Tobit, the standard deviation of the latent errors. For non-normal distributions, it is a scaling factor that gets multipled by the standard deviation for the relevant latent errors distribution. (Strictly speaking, this latter interpretation always holds, but since the standard deviation for the standardized normal is 1, we get the simple interpretation in the normal error case.)

The standard error of the regression is an analogue to the same concept in a linear regression. Defining the residuals as the difference of the observed and the expected value of the observed, we compute the standard deviation of the residuals (d.f. corrected using the number of mean coefficients).

Roughly speaking, the scale factor is related to the variability of the errors in the latent space while the standard error of the regression is the variability of the errors in observed space, given the ML estimate of the model. The estimate of the scale factor is via ML. The standard error of the regression estimate is a simple moment estimator that may or may not have great properties.

Thanks for the, as usual, clear and helpful explanation.