The reason that cables are rated for a particular current is due to thermal effects; a cable carrying a current gets warm, and the NEC tables assume a given load for a given temperature rise, and make assumptions about average loadings or derate for continuous loads.
Welders are generally only used for a small duty cycle, often just 20%, meaning something like 5 minutes welding, 15 minutes not-welding.
The upshot of this is that the conductors have a faster than expected rate of temoperature rise, as more current is flowing through a given conductor than normal calcs would allow. But... then the conductors get plenty of time to cool down. Thus the duty cycle of the application is factored in to the cable behaviour allowing you to use less heavy cable than you would normally for that size load and thus breaker size.
This leeway is not unique; POCOs allow terrible overloads to be placed on their transformers, particularly when there has been a contingent fault somewhere nearby, but they do this knowing that the tranny will only have to cook for a couple of hours, and then it'll be able to cool down again.