While there is not a term that I know of that specifically refers to singing "too fast" (apart from saying that someone is out of time), composers refer to how text is set as prosody. If someone clearly puts too many words into a given line, we would say that their prosody is poor or under-developed.
It is not rapping as rapping is not a sung style. The closest possible approximation would be recitative technique used by vocalists / composers in operatic and musical theater contexts in which performs sing-speak over repeated accompaniment.
Most of the rhythmic divisions used by the artist in the example are duple divisions, in other words, he is subdividing by powers of two in order to make the words fit in the appropriate time. He is not changing tempo or meter. His phrasing is sometimes stressed over the barline, and some syncopation results.
But by and large, when speaking about his delivery and use / treatment of the text, you would use "prosody" to describe what you mean.