The song in question has several soloists singing at different times. In the recording mentioned, some of the soloists have a classical voice production, which you typically hear in vocal works, but a few of them have a voice production that I can't quite characterize.
You can hear the voices in question starting at 1:25s. The first, female, vocal I would describe as shouty, intense and choked. The second, male, vocal has the approximately the same voice production, I think, but without the intensity. The third, male, vocal sounds classical and provides a defining contrast to the first two.
How do you "do" the female vocal at about 1:28s ? Is it a matter of voice character and anatomy, or can it be trained? Is it much more different than classical singing? Can you add some better known songs in which the vocal production is similar?