It is called Scat singing.
Per Wikipedia:
In vocal jazz, scat singing is vocal improvisation with wordless vocables, nonsense syllables or without words at all.
It is pretty common in Jazz. Singers used it to 'solo' (cause they had no instrument).The song you provided is jazzy so I think scat singing is the correct term here.
Here are a few examples:
and
This falls under the 'wordless vocals' category:
Not exactly scatting, but they are both wordless vocals.