My 120A welder has a rating plate of needing 26A at full 120A welding current, but you only use that kind of current with a 6mm welding rod ( quarter inch in the USA) to weld a pair of 30mm steel plates together, and if you are doing that at all regular then you need a bigger welder in any case. Open circuit voltage is around 70VAC, but drops to around 10VAC when you have an arc struck, so running it off a standard 16A socket is fine. Typically I use it around 60A or so, with 2.5mm rods, and there it is fine.
Of course, now I have an inverter welder I use it instead most of the time, much better weld, much better starting and I can actually weld thin plate with it, instead of having an impromptu plasma cutter that uses a welding rod. But yes, the welder transformer puts the power from the wall output into a very small spot, so can do some serious heating on metal.