The traces are extremely short. Look at a modern motherboard and you'll find a bank of capacitors and regulators about 2cm away from the CPU socket.

If you've got 4 layers of 2oz copper, and you make the positive and negative traces 10mm wide, you'll only be dissipating 28 watts when the CPU is dissipating 300 watts. And most motherboards have more than 4 layers and have space for more than 10mm of power trace width. And there's a bunch of forced air cooling, due to that 300 watts of heat the CPU is producing.

Electrical code doesn't let buildings use cables that dissipate 28 watts for 2cm of distance because it would be extremely problematic if your 3m long EV charge cable dissipated 4200 watts.

That code is for round wire (minimal surface area per volume) that can be placed inside insulation in walls.

This 350A is flat conductors (maximal surface area thus heat dissipation) and very short (not that much power to dissipate so the things it connects to have a significant effect on heat dissipation).

Those electrical code cross sections are for 350A at 230V, corresponding to about 80 kW (400V is the same as it's actually three 230V wires).

Processors operate at about 1V. At 300W it's enough to use a much smaller cross section, which is split across many traces.

That's not how I^2R losses work. Voltage is not relevant.

I agree. However voltage is relevant for insulation, which also affects how heat can dissipate for the wire, and might also be relevant for a failing wire, when higher and higher voltage can build up at the point of failure (not sure if it's a common engineering consideration outside of fuses, which are designed to fail).

At 300W there's only so much power that can become heat. With 80 kW flowing through the wire, if the insulation melts due to excessive wire resistance you call the fire brigade.

Bursty current spikes, short and fat traces, using the motherboard as a heat sink, active cooling, and allowing the temperature to rise quite a bit. If you look at thermal camera videos[0], it pretty clear where all the heat is going (although a significant part of that is coming from the voltage regulators).

On the other hand, your national electrical code is going to assume you're running that 350A cable at peak capacity 24/7, right next to other similarly-loaded cables, stuffed in an isolated wall, for very long runs - and it still has to remain at acceptable temperatures during a hot summer day.

[0]: https://www.youtube.com/watch?v=YyDMlXEZqb0