I don’t think it could have been one drink. I think a series of lets say 8-10 drinks were the culprit. You can probably drink one drink that is -60f but not 6 in a row. Something like that.
I don’t know what the temperature of the drink was but I agree that I don’t think you could reliably drink liquid nitrogen. If it managed to get to your esophagus I think you would either rupture something internally or be belching because of the expanding gas.
looking at that photo of the drink itself, it appears that it might be using ‘dry ice’ which is a frozen solid of nitrogen.
a tiny block of dry ice would make that cold smoke on the top of the drink, and you could easily swallow the piece of dry ice (not good!)