It is a generalization of the geometric median or 1-median algorithm, defined for a single cluster.
The sum of distances is widely used in applications such as the facility location problem.
This makes the algorithm more reliable for discrete or even binary data sets.
In contrast, the use of means or Euclidean-distance medians will not necessarily yield individual attributes from the dataset.
However, a medoid has to be an actual instance from the dataset, while for the multivariate Manhattan-distance median this only holds for single attribute values.