How is the standard error of the estimate similar to that of standard error terms for t-tests?

both terms provide a measure of between treatments variances from sample means or predicted values.

the two terms are not conceptually similar.

both terms provide a measure of the treatment effects or accuracy of prediction, such that higher values of standard error are more likely to lead to significant results.

either term provides a measure of how much difference to expect between either sample means or predicted values and observed values.