So how did the English language expression "sweating” like a pig" develop?

It's actually a reference to pig iron, which is form of iron smelting:
When pig iron is originally created from iron ore, the smelter needs to heat the ore to extreme temperatures, and then move the liquid metal into the mold. Until the liquid cools, it can't be safely moved, as the extremely hot metal is liable to spill, burning whatever it comes in contact with.

How does the smelter know when the metal is cool enough to transport? When the "pigs" sweat" As the metal cools, the air around it reaches the dew point, causing droplets to form on the metal's surface.