First of all, I don’t think this is the right forum to ask such type of questions, since they are not specifically related to PyTorch.
Anyway, I think they call it multi-class loss to distinguish it from the binary classification loss, which is usually written as something like:

where the y_i’s are the binary labels (either 0 or 1) and the p_i’s are the outputs of your binary classifier (usually, the output of a sigmoid). It is easy to see that the multi-class loss may be reduced to this loss when K=2, by doing a simple reparameterization that transforms a binary softmax into a sigmoid.