Although the photographer of the example images has categorized the samples you show as "documentary", they're certainly not straight documentary. They're probably not really even properly called portraitsportraits; they might be called caricatures.
But you ask what purpose a photographer might have in making photographs in this way, and that I think is straightforward: the photographer wants to make something that looks interesting and different, rather than yet another example of the same thing over again. We may argue about whether it's successful, but the purpose of this approach is clearly to make a photograph that looks different from the expected.
In fact, in this particular series, the distortion appears to have a specific artistic purpose; in addition to the "documentary" and "street" tags, these are labeled as "conceptual", and are part of a small series provocatively titled "An Enemy of the People". (To see all seven, start with this one and then click next.) It seems very likely that the literal strongly-distorted views of ordinary people on the street is an intentional statement related to that title.
So, while the purpose of a distorted-perspective street shot may simply be humor, or nothing more significant than a way to force a new perspective onto an old scene, it can certainly be used in other ways as part of a photographic vocabulary.