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I know that the standard error is calculated from looking at the sum of squared errors divided by square root of n.
However, as we add more and more variables, we expect the sum of squared errors to decrease and eventually overfit. So I expect to see the standard error to decrease with each variable on the anova table but it doesn't.
Is it fitting a single variable each row or is the error calculated from the prediction with all previous rows too?
That looks like a table of jointly estimated coefficients, not an anova table.
This means the standard errors are all about the same value. You can also interpret the wald statistic and p-value as being "marginal" tests/effects. That is, if that predictor was last in a "sequential" anova table.
