There would generally (though obviously not always) be some wheel slide in conjunction with a stoppie. That would mean the wheel goes from a few Kmh to zero, with no transition, which should trigger the ABS. In checking around I've found that wheel lift is detected in rain and power modes of the ABS, but not in race or slick modes. That's why this could happen. As I mentioned earlier, he was likely in race mode.
As far as I understand, you can pull a stoppie without ever locking any of the wheels (unlike in the OP video). So, under that assumption, ABS never gets in the action to identify what's a stoppie, let alone prevent it.
Switching to Rain mode indeed activates the ABS, but I think it also activates some gravitational sensors both front and back in order to prevent stoppies and wheelies.