Switch (console only, without joycons) :
23.9 cm (width) by 10.2 cm (height)
(9.41 in × 4.02 in)
= 243.78 cm²
PS Vita (1st model, wifi version) :
18.2 cm (width) by 8.35 cm (height)
(7.2 in x 3.289 in)
= 151.97 cm² (even a little less, vita is not rectangular)
I put my ps vita with a soft case or my 3DS in my pocket, but ps vita is close to the maximum size I can put, so I know I can't do the same with the Switch.
So yes, Switch is bigger, and his claim sounds legit to me.