When can AUC and accuracy rate be equal?
AUC (or most often AUROC = "area under receiver operating characteristic") and accuracy are different measures, but used for same purpose - to objectively measure performance of a simple binary classifier.

The two measures can be equal at extreme values of 0 and 1 for perfect classifiers - or inverse perfect classifiers (you can just invert the output to get a perfect classifier). It might be possible for the values to be numerically equal at other times, but if so it would be a coincidence with no specific meaning.

Both these metrics can be used with a simple classifier that only outputs true or false values for class membership. However, the AUROC metric requires some kind of parameter you can vary in order to plot the ROC curve. Usually this is a threshold for classification, used against the class probability output of a classifier.

There are other possible metrics. For example, F1 score and cross entropy. The F1 score again has 1.0 for a perfect classifier and 0.0 for a bad classifier, but cross entropy scores lower for better classifiers - 0.0 for perfect and no upper limit for bad output. Again, the values of these might be equal to other metrics at some points, but if so it is not meaningful.

If you are comparing two classifiers for a particular task, then it is important to compare them using the same metric, on the same test data. The metric and test data you choose should relate to your original problem.

If you are reading other people's published results, and want to compare them, it is not really possible if one person has used AUC and the other accuracy.